Design for fluctuating loads

8/19/2019 Design for fluctuating loads

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Department of Mech

ESIGN AGAINST FLUCTUATING

Don Bosco Co

A group

NIRAJ KAKATI

RUPJYOTI BAR

FEROZ AHMED

RIJUAN HUSSA

nical engineering, School of Technology


LOA S

ege of Engineering & technology, Guwahati

presentation report on ACHINE DESIGN

Submitted by-

DC2013BT

UAH DC2013BT

MAZUMDER DC2013BT

IN DC2013BT

6th

semester - 2016

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, ASSAM DON BOSCO

UNIVERSITY


Assam)

ACHINE DESIGN

0142

0263

0017

0134


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P a g e | 2

Department of Mechanical engineering, School of Technology, ASSAM DON BOSCO

UNIVERSITY

Sl no. Contents Page no.

1. Declaration i

2. Certificate ii-iii

3. Abstract iv

4. Acknowledgement v

Sl no.

ContentsPage no.

1. Introduction 1

2. Stress concentration 2-4

3. Reduction of Stress concentration factor 5-8

4. Fluctuating stresses 9-10

5. Fatigue failure 11-12

6. Endurance limit 13-18

7. Low cycle and high cycle fatigue 19

8. Notch sensitivity 20-21

9. Soderberg & Goodman lines 22-25

10. Gerber line 26

11. Conclusion 27

12. Reference 28


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Department of Mech

EPARTMENT OF MECHANICAL ENGINEERING

We hereb

AGAINST FLUCTUATIN

Engineering at Don Bosco S

the requirement for the Prese

done by us under the guidanc

------------------------------

FEROZ AHMED MAZU

DC2013BTE0017

------------------------------

NIRAJ KAKATI

DC2013BTE0142



(SCHOOL OF TECHNOLOGY)

ASSAM ON BOSCO UNIVERSITY

GUWAHATI – 781017

ECLERATION

declare that the dissertation work e

STRESSES” submitted to the Departm

hool of Technology, Guwahati, Assam, in

ntation Seminar in MACHINE DESIGN,

of Sir. TAPAS KALITA.

---------------

DER

----------------------------

RI

D

---------------

-----------------------------

RUPJ

D

P a g e | 3

, ASSAM DON BOSCO

UNIVERSITY




ntitled “DESIGN

ent of Mechanical

artial fulfilment of

s an original work

-------------------

JUAN HUSSAIN

2013BTE0134

-------------------

YOTI BARUAH

2013BTE0263


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Department of Mech

EPARTMENT OF

“DESIGN AGAINST F

HUSSAIN (DC2013BTE

AHMED MAZUMDER

(DC2013BTE0263 ), as a

engineering 6 th semester ,

out by them under my su

-------------------------------

Sir. TAPAS KALITA,

Assistant Professor, Mechanical En

School of Technology,

Assam Don Bosco University





GUWAHATI – 781017

CERTIFICATE

his is to certify that the presentatio

LUCTUATING STRESSES” submi

134 ), NIRAJ KAKATI (DC2013BT

(DC2013BTE0017 ), and RUPJY

assignment for MACHINE DESIG

is a bona fide record of the presentat

ervision during the year 2016.

----------------------

gineering

P a g e | 4

, ASSAM DON BOSCO

UNIVERSITY

MECHANICAL ENGINEERING



Report entitled

tted by RIJUAN

0142 ), FEROZ

TI BARUAH

in Mechanical

on work carried


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Department of Mech

EPARTMENT

This isAGAINST FLUCTUATIN

MAZUMDER-(017), R

RIJUAN HUSSAIN-(13

of the degree of Bachelo

Don Bosco University, G

carried out by them durin

------------------------------

Dr. Mrs. LEENA H.

Head Of Department, Mechanical E

School of Technology,

Assam Don Bosco University





GUWAHATI – 781017

CERTIFICATE

to certify that the seminar project e G STRESSES” is submitted by F

PJYOTI BARUAH-(263), NIRAJ

) are the students of 6 th

Semester , in p

of Technology in Mechanical Engin

wahati is a bonafide record of their p

the academic year 2016.

-------------------

NEMADE

ngineering

P a g e | 5

, ASSAM DON BOSCO

UNIVERSITY

OF MECHANICAL ENGINEERING



titled “DESIGNROZ AHMED

AKATI-(142),

artial fulfillment

ering of Assam

esentation work


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Department of Mech


The pu

considerations required

work is based on the Ma

such designs the principl

loads where the compo

magnitudes with time. It

components are due to '

practice, the pattern of stof stresses due to vibrati

for stress-time relationshi

Keywords: stress co

concentration, fluctuatin

gerber equations and mo





GUWAHATI – 781017

ABSTRACT

rpose of our project is to study and ex

uring Design against Fluctuating lo

chine Design which works under fluc

s are based according to various cycli

ents subjected to forces are not sta

is observed that about 80% of failur

atigue failures' resulting from fluctu

ess variation is irregular and unpredins. For the purpose of design analysi

ps are used.

centration and stress factor, redu

stresses, endurance limit, soderberg,

ified goodman diagrams.

P a g e | 6

, ASSAM DON BOSCO

UNIVERSITY




lain the various

ds. Our project

uating loads. In

and continuous

tic, but vary in

s of mechanical

ting stresses. In

table, as in case, simple models

ction of stress

goodman lines,


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Department of Mech


ACKNOWLE GEMENT

It gave

our group project repoHowever, it would not h

many individuals. We w

specially our group mem

teacher, Sir. Tapas Kasupport as well as forwhich really helped us in

gratitude toDr. Manor(Head of Department, M

at the Don Bosco School

Our sincere thanks and a

their views in developinout with their abilities.

Lastly,

for helping us out in ou

work.

Thank You,

With regards,

Rupjyoti Baruah- (DC20

Niraj Kakati- (DC2013B

Rijuan Hussain- (DC201

Feroz Ahmed Mazumde





GUWAHATI – 781017

ACKNOWLE GEMENT

us immense pleasure and satisfaction

t as a reflection of our hard worve been possible without the kind sup

ould like to extend our sincere thank

ers. We are really very much thankf

ita , (Asst. Professor MNE) for h

roviding necessary information regcompleting our project report. We ext

anjan Kalita(Principal)

Mrs. Leechanical Dept.) and other teachers o

f Technology for their valuable supp

preciations are also for our class-mat

the report and people who have wil

e would like to thank everybody an

hard times throughout the group pr

13BTE0263)

E0142)

3BTE0134)

- (DC2013BTE0017)

P a g e | 7

, ASSAM DON BOSCO

UNIVERSITY




in giving away

and sincerity. port and help of

s to all of them

l to our subject

s guidance and

rding the topicend our heartfull

a H. Nemadethe department

rt and guidance.

s for lending us

lingly helped us

d The Almighty

sentation report


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1. INTRODUCTION

MACHINE DESIGN- Machine design is defined as the use of scientific principles,

technical information and imagination in the description of a machine or a mechanical system

to perform specific functions with maximum economy and efficiency.

In designing a machine component, it is necessary to have a good

knowledge of many subjects such as Mathematics, Engineering Mechanics, Strength of

Materials, Theory of Machines, Workshop Processes and Engineering Drawing. A machine

element, after design, requires to be manufactured to give it a shape of a product. Therefore,

in addition to standard design practices like, selection of proper material, ensuring proper

strength and dimension to guard against failure, a designer should have knowledge of basic

manufacturing aspects. First and foremost is assigning proper size to a machine element from

manufacturing view point. As for example, a shaft may be designed to diameter of, say, 40

mm. This means, the nominal diameter of the shaft is 40 mm, but the actual size will be

slightly different, because it is impossible to manufacture a shaft of exactly 40 mm diameter,

no matter what machine is used. In case the machine element is a mating part with another

one, then dimensions of both the parts become important, because they dictate the nature of

assembly. The allowable variation in size for the mating parts is called limits and the nature

of assembly due to such variation in size is known as fits.

=

; ,

=

; , =

The above equations are called elementary equations. A plate with a

small circular hole, subjected to tensile stress and the distribution of stress near the hole can

be seen by photo-elastic techniques. Here, we specially discuss about fatigue failure which

results due to fluctuating stresses.

The majority of engineering failures are caused by fatigue. Fatigue

failure is defined as the tendency of a material to fracture by means of progressive brittle

cracking under repeated alternating or cyclic stresses of intensity considerably below the

normal strength. Although the fracture is of a brittle type, it may take some time to propagate,depending on both the intensity and frequency of the stress cycles. Nevertheless, there is very

little, if any, warning below failure if the crack is not noticed. The number of cycles required

to cause fatigue failure at a particular peak stress is generally quiet large, but it decreases as

the stress is increased. For some mild steels, cyclical stresses can be continued indefinitely

provided the peak stress (sometimes called fatigue strength) is below the endurance limit

value. A good example of fatigue failure is breaking a thin steel rod or wire with your hands

after bending it back and forth several times in the same place. Another example is an

unbalanced pump impeller resulting in vibrations that can cause fatigue failure.


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2. STRESS CONCENTRATION

Mathematical analysis and experimental measurement show that in a

loaded structural member, near changes in the section, distributions of stress occur in whichthe peak stress reaches much larger magnitudes than does the average stress over the section.

This increase in peak stress near holes, grooves, notches, sharp corners, cracks, and other

changes in section is called stress concentration. The section variation that causes the stress

concentration is referred to as a stress raiser . In order to consider the effect of stress

concentration and find out the localised stresses, a factor called stress factor is used. There

are many reasons for rise of stress concentration.

Locally high stresses can arise due to-

–Abrupt changes in section properties (hole, corner)

–Contact stresses (bearing, gear, etc)

–Material discontinuities

–Initial stresses due to manufacturing process

– Cracks

Stress concentration is defined as the localization of high stresses due

to the irregularities present in the component and abrupt changes of the cross section. In the

figure below it can be seen the stress concentration at the corners, due to change in the

dimensions.

Structure is often designed without considering them followed by

local fixes. Instead we often assume some defects of given size as safety measure.

Geometrical features such as notches and corners give rise to stress concentrations. In

industrial components these features are often designed with a constant radius, however it is

already known that a more complex shape, having a variable radius, can have a much lower

stress concentration factor. In this paper we describe two new approaches for obtaining usefulvariable-radius notches. The first approach, which we call the Local Curvature Method


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(LCM) involves post-processing results of a stress analysis conducted on a constant-radius

notch, altering the local curvature as a function of the local surface stress. This method is

being described here for the first time: it was found to be very successful, reducing the

maximum stress at a 90o fillet by about a factor of 2. The second approach involved using

commercial software (mode Frontier) to carry out a more systematic search of possiblevariable-radius designs using multiple finite element models. This approach, though much

more expensive in terms of computing resources, was able to find slightly better solutions.

Our findings were verified by conducting experimental tests to measure brittle fracture

strength and high-cycle fatigue strength.

The below figure with a hole in middle section can have a high value of

stress concentration and it can be found out by following equations- (K c= stress concentration

factor).

Engineering components and structures almost invariably containregions of high local stress, created by a combination of geometry and loading. For

convenience we will refer to these stress concentration features as “notches”, though the work

described here is applicable to all geometric features, including holes, corners, bends and

keyways. Such features are normally designed with constant radii. The resulting stress

concentration factor K t or K c is a function of the notch radius ρ as well as other parameters

related to the component geometry and loading. However, it has already been established that

the constant radius notch is not the best solution variable radius notches, defined as notches in

which the radius varies from place to place along the notch surface, can have much lower K t

factors.

STRESS CONCENTRATION FACTOR

Stress concentration factor can be defined as the ratio of Highest value

of actual stress near discontinuity to the nominal stress obtained by elementary equations for

minimum cross- section. It is denoted by K t or K c.

The causes of stress concentration are as follows-

1. Variation in properties of materials.

2.

Load application.3. Abrupt changes in cross sections

4. Discontinuity in the component

5.

Machining scratches.


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Below figure shows a large plate that contains a small circular hole. For

an applied uniaxial tension the stress field is found from linear elasticity theory. In polar

coordinates the azimuthal component of stress at point P is given as-

The maximum stress occurs at the sides of the hole where ρ = r and θ = 12π or

θ = 3 2π . At the hole sides,

σ θ = 3σ

fig- Infinite plate with a small circular hole.

The peak stress is three times the uniform stress σ . To account for

the peak in stress near a stress raiser, the stress concentration factor or theoretical stress

concentration factor is defined as the ratio of the calculated peak stress to the nominal stress

that would exist in the member if the distribution of stress remained uniform that is-

The nominal stress is found using basic strength-of-materials

formulas, and the calculations can be based on the properties of the net cross section at the

stress raiser. Sometimes the overall section is used in computing the nominal stress. If σ is

chosen as the nominal stress for the case shown in Fig., the stress concentration factor is-

The effect of the stress raiser is to change only the distribution of

stress. Equilibrium requirements dictate that the average stress on the section be the same in


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the case of stress concentration as it would be if there were a uniform stress distribution.

Stress concentration results not only in unusually high stresses near the stress raiser but also

in unusually low stresses in the remainder of the section. When more than one load acts on a

notched member (e.g., combined tension, torsion, and bending) the nominal stress due to each

load is multiplied by the stress concentration factor corresponding to each load, and theresultant stresses are found by superposition. However, when bending and axial loads act

simultaneously, superposition can be applied only when bending moments due to the

interaction of axial force and bending deflections are negligible compared to bending

moments due to applied loads.

We can reduce the stress concentration by many ways like, additional

notches, fillet radius, drilling in the shafts, etc.

3. REDUCTION OF STRESS CONCENTRATION

A qualitative discussion of techniques for avoiding the detrimental effects

of stress concentration is given by Layer. As a general rule, force should be transmitted from

point to point as smoothly as possible. The lines connecting the force transmission path are

sometimes called the force (or stress) flow, although it is arguable if force flow has a

scientifically based definition. Sharp transitions in the direction of the force flow should be

removed by smoothing contours and rounding notch roots. When stress raisers are

necessitated by functional requirements, the raisers should be placed in regions of low

nominal stress if possible.

It is well known in contact problems that a stress singularity exists at the

sharp corner of a wedge indenter compressing a semi-infinite body. The power of the

singularity depends on the angle of the wedge and on the friction between the indenter and

the body. The investigations reported here describe how the singularity (stress concentration)

can be controlled and eliminated by relief cuts (notches) either along the free edges of the

indenter or along the interface edge of the half-plane. An indenter with fight angle corners is

investigated. Research on this problem is motivated by the difficulties associated with the

design of the upper end of an engine connecting rod, where the rod is shrunk on the wrist pin.

A related, purely asymmetric case, is that of shrink fit of a turbine disk on a shaft. The use of

notches (circular grooves) in the shaft as a stress relief device is known in the steam turbine

industry.' Although the current tests are done with plane-stress conditions, the results shed

light on the related asymmetric case.


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When notches are necessary, removal of material near the notch can

alleviate stress concentration effects, where removal of material improves the strength of the

member. A type of stress concentration called an interface notch is commonly produced

when parts are joined by welding and interface notches and one way of mitigating the effect.

The surfaces where the mating plates touch without weld metal filling, form what is, in effect,a sharp crack that causes stress concentration. Stress concentration also results from poor

welding techniques that create small cracks in the weld material or burn pits in the base

material.


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Through various experiments the stress concentration can be predicted. Some of them are-

1.

Brittle coatings

2.

Photo elasticity

3.

Thermo elasticity

4.

Strain gages.

Brittle Coating Technique: A brittle coating is sprayed on the surface and allowed to

dry. Crack patterns developed by the loading and their relation to a calibration coating

indicate regions and magnitudes of stress concentrations.

Photo elasticity Technique: A specimen with identical geometry to the actual

notched part is made of a certain transparent material. Changes in optical properties of the

transparent material under load, measured by a polar scope, indicate stress distributions and

magnitudes.

Thermo elasticity Technique: Stress distribution is obtained by monitoring small

temperature changes of the specimen or component subjected to cyclic loading.

Electrical Resistance Strain Gage: The most common experimental

measurement technique A strain gage is bonded to the surface in the region of interest.

Applied load causes dimensional changes of the gage resulting in changes to electrical

resistance, which in turn indicates the existing strain.

REDUCTION OF STRESS CONCENTRATIONS-

(i) Additional notches and holes in tension member:- A flat plate with V-

notch subjected to tension force, where a single notch shows high value of stress

concentration. This can be removed by- Use of multiple notches, drilling additional

holes, removal of undesired materials. In these three ways, the sharp bending of a

force flow line can be reduced and it follows a smooth curve, as shown in figure.


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(ii) Fillet radius, undercutting and notch for members in bending:- A bar

of circular cross- section in which its shoulder creates a change in cross section of the

shaft, which results in stress concentration. There are three methods to reduce stress

concentration at the base of this shoulder. The fillet radius results in gradual transition

from small diameter to a larger diameter. The increase in undercutting the shoulderalso can reduce the stress concentrations. A notch above the bigger diameter shaft can

also results in reduction of stress concentrations.

(iii) Drilling additional holes for shaft:- A transmission shaft with a key way is adiscontinuity and results in stress concentrations. In case of fillet radius in the innercorner of shaft, drilling two symmetrical holes on the sides of the keyway can reduce

stress concentrations.

(iv) Reduction of stress concentrations in threaded members:- A threaded

member with thread joins is shown that the force flow lines are more at the bent as it

passes from the shank portion to threaded portion of the component. This results in

stress concentrations. A small undercut between the shank and the threaded portion

can reduce the stress concentrations. Or we can reduce the shank diameter equal to the

core diameter of the thread. In such case the force flow line is almost straight an there

is no stress concentrations.


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4. FLUCTUATING STRESSES

In many applications of designs, the stresses are based according to

various cyclic and continuous loads where the components subjected to forces are not static, but vary in magnitudes with time. It is observed that about 80% of failures of mechanical

components are due to 'fatigue failures' resulting from fluctuating stresses. In practice, the

pattern of stress variation is irregular and unpredictable, as in case of stresses due to

vibrations. For the purpose of design analysis, simple models for stress-time relationships are

used. The most popular model for stress-time relationship is the sine curve. A typical stress

cycle is shown in figure below where the maximum, minimum, mean and variable stresses

are indicated. The mean and variable stresses are given by,

σmean= mean stress; σvariable= mean amplitude

fig- A typical stress cycle showing maximum, mean and variable stresses.

Conditions often arise in machines and mechanisms when stresses fluctuate

between a upper and a lower limit. For example in figure below, the fibre on the surface of arotating shaft subjected to a bending load, undergoes both tension and compression for each

revolution of the shaft.

figure- Stresses developed in a rotating shaft subjected to a bending load.


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Any fibre on the shaft is therefore subjected to fluctuating stresses.

Machine elements subjected to fluctuating stresses usually fail at stress levels much below

their ultimate strength and in many cases below the yield point of the material too. These

failures occur due to very large number of stress cycle and are known as fatigue failure.

These failures usually begin with a small crack which may develop at the points ofdiscontinuity, an existing subsurface crack or surface faults. Once a crack is developed it

propagates with the increase in stress cycle finally leading to failure of the component by

fracture. There are mainly two characteristics of this kind of failures:

(i)

Progressive development of crack.

(ii) Sudden fracture without any warning since yielding is practically absent.

Fatigue failures are influenced by-

a)

Nature and magnitude of the stress cycle.

b)

Endurance limit.

c)

Stress concentration.

d)

Surface characteristics.

These factors are therefore interdependent. For example, by grinding

and polishing, case hardening or coating a surface, the endurance limit may be improved. For

machined steel endurance limit is approximately half the ultimate tensile stress.

There are 3 types of mathematical models for cyclic stresses :-

1.

Fluctuating stresses or Alternating stress.2. Repeated stresses.

3. Reversed stresses.

The fluctuating stresses varies in a sinusoidal manner with respect to

time. It has some mean value as well as amplitude values. It fluctuates between maximum

and minimum values of stress. The stresses which vary from a minimum value to a maximum

value of the same nature, (i.e. tensile or compressive) are called fluctuating stresses.

Repeated stress also varies in a sinusoidal manner with respect to time, but the variation is

from zero to some maximum values. The stresses which vary from zero to a certain

maximum value are called repeated stresses. The minimum stress is zero in this case andtherefore, amplitude stress and mean stress are equal. The stresses which vary from a

minimum value to a maximum value of the opposite nature (i.e. from a certain minimum

compressive to a certain maximum tensile or from a minimum tensile to a maximum

compressive) are called alternating stresses. The stresses which vary from one value ofcompressive to the same value of tensile or vice versa, are known as completely reversed or

cyclic stresses. Reversed stress also varies in a sinusoidal manner with respect to time, but it

has zero mean stress. In case, half portion of the cycle consists of tensile stress and the

remaining half of compressive stress. There is a complete reversal from tension to

compression between these two halves and therefore, the mean stress is zero. The below

figure explains the three stresses.


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In the analysis of fluctuating stresses, tensile stress is considered as positive, while compressive stress as negative.

5. FATIGUE FAILURE

Often, machine members are found to have failed under the action of

repeated or fluctuating stresses; yet the most careful analysis reveals that the actual maximum

stresses were well below the ultimate strength of the material, and quite frequently even

below the yield strength. The most distinguishing characteristic of these failures is that the

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

failure. When machine parts fail statically, they usually develop a very large deflection,

because the stress has exceeded the yield strength, and the part is replaced before fracture

actually occurs. Thus many static failures give visible warning in advance. But a fatigue

failure gives no warning! It is sudden and total, and hence dangerous. It is relatively simple to

design against a static failure, because our knowledge is comprehensive. Fatigue is a much

more complicated phenomenon, only partially understood, and the engineer seeking

competence must acquire as much knowledge of the subject as possible.

In narrow sense, the term fatigue of materials and structural

components means damage and damage due to cyclic, repeatedly applied stresses. In a wide

sense, it includes a large number of phenomena of delayed damage and fracture under loads

and environmental conditions. It is expedient to distinguish between high-cycle (classic) and

low-cycle fatigue. Plastic deformations are small and localized in the vicinity of the crack tip

while the main part of the body is deformed elastically, then one has high-cycle fatigue. If the

cyclic loading is accompanied by plastic deformation in the bulk of the body, then one has a

low-cycle fatigue. Usually we say low-cycle fatigue if the cycle number up to the initiation of

a visible crack or until final fracture is below 104 or 5.10

4 cycles.

Fatigue fracture typically occurs in material of basically brittle nature.External or internal cracks develop at pre-existing flaws or fault of defects in the material


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these cracks then propagate and eventually they lead to total failure of part. The fracture

surface in fatigue is generally characterized by the term “beach marks”.

It is expedient to distinguish between high-cycle (classic) and low-

cycle fatigue. Plastic deformations are small and localized in the vicinity of the crack tipwhile the main part of the body is deformed elastically, then one has high-cycle fatigue. If the



a visible crack or until final fracture is below 104 or 5.104 cycles.

A fatigue failure has an appearance similar to a brittle fracture, as the

fracture surfaces are flat and perpendicular to the stress axis with the absence of necking. The

fracture features of a fatigue failure, however, are quite different from a static brittle fracture

arising from three stages of development.

Stage I is the initiation of one or more micro cracks due to cyclic plastic deformation

followed by crystallographic propagation extending from two to five grains about the origin.

Stage I cracks are not normally discernible to the naked eye.

Stage II progresses from micro cracks to macro cracks forming parallel plateau-like fracture

surfaces separated by longitudinal ridges. The plateaus are generally smooth and normal to

the direction of maximum tensile stress. These surfaces can be wavy dark and light bands

referred to as beach marks or clamshell marks, as seen in Fig. 6–1. During cyclic loading,

these cracked surfaces open and close, rubbing together, and the beach mark appearance

depends on the changes in the level or frequency of loading and the corrosive nature of theenvironment.

Stage III occurs during the final stress cycle when the remaining material cannot support

the loads, resulting in sudden, fast fracture. A stage III fracture can be brittle, ductile, or a

combination of both. Quite often the beach marks, if they exist, and possible patterns in the

stage III fracture called chevron lines, point toward the origins of the initial cracks.


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Fatigue failure is due to crack formation and propagation. A fatigue

crack will typically initiate at a discontinuity in the material where the cyclic stress is a

maximum.

Discontinuities can arise because of:-

1.

Design of rapid changes in cross section, keyways, holes, etc. where stress

concentrations occur.

2.

Elements that roll and/or slide against each other (bearings, gears, cams, etc.)

under high contact pressure, developing concentrated subsurface contact stresses,

that can cause surface pitting or spalling after many cycles of the load.

3. Carelessness in locations of stamp marks, tool marks, scratches, and burrs; poor

joint design; improper assembly; and other fabrication faults.

4.

Composition of the material itself as processed by rolling, forging, casting,

extrusion, drawing, heat treatment, etc. Microscopic and sub microscopic surfaceand subsurface discontinuities arise, such as inclusions of foreign material, alloy

segregation, voids, hard precipitated particles, and crystal discontinuities.

Various conditions that can accelerate crack initiation include residual

tensile stresses, elevated temperatures, temperature cycling, a corrosive environment, and

high frequency cycling. The rate and direction of fatigue crack propagation is primarily

controlled by localized stresses and by the structure of the material at the crack. However, as

with crack formation, other factors may exert a significant influence, such as environment,

temperature, and frequency.

In many applications, the behaviour of a component in service is

influence by several other factors besides the properties of the material used in its

manufacture. This is particularly true for the cases where the component or structure is

subjected to fatigue loading, the fatigue resistance can be greatly influenced by the service

environment, surface condition of the part, method of fabrication and design details. In some

cases, the role of the material in achieving satisfactory fatigue life is secondary to the above

parameters, as long as the material is free from major flaws.

6. ENDURANCE LIMIT

The fatigue or endurance limit of a material is defined as the

maximum amplitude of completely reversed stress that the standard specimen can sustain for

an unlimited number of cycles without fatigue failure. It is the fatigue strength of a material.

Fatigue strength is defined as the maximum stress that can be endured

for a specified number of cycles without failure. Low cycle fatigue strength approaches the

static strength. When the cycle number exceeds to one limit, the fatigue strength falls to

fraction of the static strength. The fatigue strength is the value of the alternating stress that


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results in failure by fracture a specific number of cycles of load application. It can also be the

ordinate of the σ-n (stress versus number of cycles to failure) curve. The fatigue behaviour of

a specific material, heat treated to a specific strength level is determined by a series of

laboratory tests on a large number of apparently identical samples of those specific materials.

The specimens are machined with shape characteristics which

maximize the fatigue life of a metal, and are highly polished to provide the surface

characteristics which enable the best fatigue life. A single test consist of applying a known,

constant bending stress to a round sample of the material, and rotating the sample around the

bending stress axis until it fails. As the sample rotates, the stress applied to any fibre on the

outside surface of the sample varies from maximum-tensile to zero to maximum compressive

and back. The test mechanism counts the number of rotations (cycles) until the specimen

fails. A large number of tests is run at each stress level of interest, and the results are

statistically massaged to determine the expected number of cycles to failure at that stress

level. The cyclic stress level of the first set of tests is some large percentage of the Ultimate

Tensile stress (UTS), which produces failure in a relatively small number of cycles.

Subsequent tests are run at lower cyclic stress values until a level is found at which the

sample will survive 10 million cycles without failure. The cyclic stress level that the material

can sustain for 10 million cycles is called the Endurance limit (EL).

FATIGUE FAILURE TESTING-

The R. R. Moore’s high-speed rotating beam machine is used to

performing a fatigue test. The specimen, shown in Fig. below, is rotating with a constantangular speed. The specimen is subjected to pure bending by means of weights. The intensity

of the reversed stress causing failure after a given number of cycles is the fatigue strength

corresponding to that number of loading cycles. The fatigue strengths considered for each test

are plotted against the corresponding number of revolutions. The resulting chart is called the

strength-life, S-N, diagram. The diagram depicts the fatigue strength versus cycle life N of a

part. The S-N curves are plotted on log-log coordinates.


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Suppose that a particular specimen is being fatigue tested. Now

suppose the fatigue test is halted after 20% to 25% of the expected life of the specimen, and

the surface condition is restored to its original state. Now the fatigue test is resumed at the

same stress level as before. The life of the part will be considerably longer than expected. If

that process is repeated several times, the life of the part may be extended by several hundred percent, limited only by the available cross section of the specimen. That proves fatigue

failures originate at the surface of a component.

A failure that results from such cyclic loads is called a fatigue

failure. Since many structural components are subjected to cyclic loads it is necessary for the

design engineer to have some quantitative measure of the material’s ability to withstand such

repeated loads. Quantitative data for the fatigue properties of a given material are obtained by

subjecting a number of standard specimens to cyclic loads until fracture occurs. (Joseph

Datsko, 1997) The objective of the fatigue strength or fatigue limit test is to estimate a

statistical distribution of the fatigue strength at a specific high-cycle fatigue life. Amongmany fatigue strength tests methods, the staircase method (often referred as the up-and-down

method) is the most popular one that has been adopted by many standards to asses statistical

of a fatigue limit. In this test, the mean fatigue limit has to first estimated, and a fatigue life

test is the conducted at a stress level a little higher than the estimated mean. If the specimen

fails prior to the life of interest, the next specimen has to be tested at a lower stress level.

Therefore, each test is dependent on the previous test results, and the test continuous with a

stress level increased or decreased.

FATIGUE DAMAGE PROCESSFatigue is gradual process of damage accumulation that proceeds

on various levels beginning from the scale of the crystal lattice, dislocations and other objects

of solid state physics up to the scales of the structural components. Three or four stages of

fatigue damage are usually distinguishable. In the first stage, the damage accumulation occurs

on the level of grains and inter granular layers. The damage is dispersed over the volume of a

specimen or structural component, or at least, over the most stressed parts. At the end of this

stage, nuclei of microscopic cracks originate, example, such aggregates of micro cracks that

are strong stress concentrators and under the following loading, have a tendency to grow.

Surface nuclei usually can be observed visually (at least with proper magnification). Thesecond stage is the growth of cracks that depth is small compared with the size of cross

section. At the same time, the sizes of these cracks whose depth is small compared with the

size of the cross section. At the same time, the sizes of these cracks are equal to few

characteristics scales of microstructure, say, to several grain sizes. Such cracks are called

small cracks. Most of them stop growing upon meeting some obstacles, but one or several

cracks transform into microscopic, “long” fatigue cracks that propagate in a direct way as

strong stress concentrators. This process forms the third stage of fatigue damage. The fourth

stage is rapid final fracture due to the sharp stress concentration at the crack front and/or the

expenditure of the material’s resistance to fracture.


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The results of the test done in the rotating beam fatigue testing

machine can be plotted in graph by means of an S-N curve. The S-N curve is a graphical

representation of stress amplitude Sf verses the number of stress cycles N before the fatigue

failure on a log-log graph paper. The S-N curve for steels is illustrated in the graph below fig.

ENDURANCE LIMIT APPROXIMATION:-

Two separate notations are used for endurance limit,

Se' = Endurance Strength of material specimen under laboratory condition

Se = Endurance Strength of material specimen under actual running condition

We have some relations for Se & Se'

Se'= 0.5 Sut (for steel)

Se'= 0.4 Sut (for CI & cast steel)

Se'= 0.4 Sut (for wrought Al alloys)

Se'= 0.3 Sut (for cast Al alloys)

These relationship is based on 50% reliability. The relation between Se' & Se -

Se = K a * K b * K c * K d * Se' Where, K a = Surface Correction factor

K b = Size Correction factor

K c =Reliability Correction factor

K d = Temperature Correction factor

Se' = Endurance Strength of material specimen under laboratory condition

Se = Endurance Strength of material specimen under actual running condition


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Surface finish factor KaThe surface finish factors in the figure are based on large number of experiments on wrought

steels and are not applicable to other ductile materials like aluminium. Cast iron is generally

not used for machine components under fluctuating loads; but if needed k a may be used as 1 because even the mirror finished. Cast-iron specimen will have graphite flakes and other

discontinuities. (K a = 1 for cast iron)

Size factor Kb for rotating circular partsThe diameter of a rotating-beam specimen is only 7.5 mm; machine part having larger

diameters are likely to have lower endurance strength than Se’ because larger the part, greater

the possibility of a flaw in the part. Therefore the chances of fatigue failure originating from

any one of those flows are more. Thus, the endurance limit of machine member is reduced by

the size factor K b as shown in the table


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Reliability Factor KcThe published fatigue strength data has a scatter, and the SN curve is plotted through the

mean of the scatter points. It means that the endurance limit has a reliability of 50%, i.e., 50-

percent components may fail earlier than the one million stress cyber. The standard deviationof the data points for steels is limited to 8% of the mean value. Using this value of standard

deviation, reliability factor K c for a desired reliably is calculated below:

K d = modifying factor for stress concentration

The endurance limit is reduced due to stress concentration and the factor used for cyclic

loading is less than the theoretical stress concentration factor due to notch sensitivity. To

reduce the endurance limit we can increasing the stress amplitude or reducing the endurance

limit. The difference between the actual increase in stress and the theoretical increase in

stress is related by ‘notch sensitivity factor q’ as follows:


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7. LOW CYCLE AND HIGH CYCLE FATIGUE

It is expedient to distinguish between high-cycle (classic) and low-

cycle fatigue. Plastic deformations are small and localized in the vicinity of the crack tip

while the main part of the body is deformed elastically, then one has high-cycle fatigue. If the



a visible crack or until final fracture is below 104 or 5.104 cycles. In material science, fatigue

is the progressive, localized, and permanent structural damage that occurs when a material is

subjected to cyclic or fluctuating strains at nominal stresses that have maximum values less

than (often much less than) the static yield strength of the material. The resulting stress may

be below the ultimate tensile stress, or even the yield stress of the material, yet still cause

catastrophic failure. A practical example of low-cycle fatigue would be the bending of a paperclip. A metal paperclip can be bent past its yield point without breaking, but repeated

bending in the same section of wire will cause material to fail.

The S-N curve above has 2 region of curve namely, high cycle and

low cycle fatigue. The difference of these two are-

1. Any fatigue failure when the number of stress cycles are less than 1000, is called-low

cycle fatigue. Any failure when the number of stress cycles are more than 1000, is called

high-cycle fatigue.

2. Failure of studs on truck wheels, failure of setscrews for locating gears on shafts or failure

of short-lived devices such as a missile is the example of low cycle fatigue. And

components like springs, ball bearings, gears etc are high cycle fatigue failures category.3. The low cycle involves fatigue with plastic yielding at the localized areas of the

components.

4. Components subjected to high cycle fatigue are designed on the basis of endurance limit

stress.


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High-cycle fatigue involves a large number of cycles (N4105 cycles)

and an elastically applied stress. High-cycle fatigue tests are usually carried out for 10 7 cycles

and sometimes 5.108 cycles for nonferrous metals. Although the applied stress is low enough

to be elastic, plastic deformation can take place at the crack tip. High cycle fatigue data are

usually presented as a plot of stress, S, versus the number of cycles to failure, N. A log scaleis used for the number of cycles. The value of stress, S, can be the maximum stress, S max, the

minimum stress, Smin, or the stress amplitude, Sa.

The S-N relationship is usually determined for a specified value of

the mean stress, sm, or one of the two ratios, R or A. The fatigue life is the number of cycles

to failure at a specified stress level, while the fatigue strength (also referred to as the

endurance limit) is the stress below which failure does not occur. As the applied stress level

is decreased, the number of cycles to failure increases. Normally, the fatigue strength

increases as the static tensile strength increases. For example, high strength steels heat treated

to over 1400 MPa (200 ksi) yield strengths have much higher fatigue strengths thanaluminium alloys with 480 MPa (70 ksi) yield strengths. A comparison of the S-N curves for

steel and aluminium. Note that steel not only has a higher fatigue strength than aluminium,

but it also has an endurance limit. Below a certain stress level, the steel alloy will never fail

due to cyclic loading alone. On the other hand, aluminium does not have a true endurance

limit. It will always fail if tested to a sufficient number of cycles. Therefore, the fatigue

strength of aluminium is usually reported as the stress level it can survive at a large total

number of cycles, usually 5·108 cycles. It should be noted that there is a considerable amount

of scatter in fatigue test results. It is therefore important to test a sufficient number of

specimens to obtain statistically meaningful results.

8.NOTCH SENSITIVITY

It is observed that the actual reduction in the endurance limit of a

material due to stress concentration is less than the amount indicated by the theoretical stress

concentration factor K t. Therefore, two separate notations, K t and K f are used for stress

concentration factors. K t is theoretical stress concentration factor, which is applicable to ideal

materials that are homogeneous, isotropic and elastic. K f is the fatigue stress concentration

factor, which is defined as-

the ratio of endurance limit of the notch free specimen to the endurance limit of the

notched specimen.

The factor K f is applicable to actual materials and depends upon grain size of

the material. It is observed that there is a greater reduction in the endurance limit of fine

grained materials as compared to coarse grained materials, due to stress concentration.

Notch sensitivity can be defined as the susceptibility of a material to succumb

to the damaging effects of stress raising notches in fatigue loadings.

The notch sensitivity is given as-

=


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actual stress= K f * σo where, σo= nominal stress

theoretical stress= K t * σo

so increase in actual over nominal stress is = K f * σo - σo

and increases in theoretical over nominal stress is = K t * σo - σoTherefore,

=Kf ∗ σo − σo

Kt ∗ σo − σo

=Kf − 1

Kt − 1 = 1 + ( − 1)

when the material has no sensitivity to notches, q=0; K f =1 and

when the material is full sensitive to notches, q=1; K f =K t

DESIGN FOR FINITE AND INFINITE LIFE TIME:-

When the components are subjected to fluctuating stresses but not

completely reversed stress, the mean stress is not zero. The stress can be purely tensile,

purely compressive or mixed depending upon magnitude of mean stress. Such problem are

solved with the help of modified Goodman diagram.

The design problem for completely reversed stress are further divided into -

1.

design for finite life .2. design for infinite life.

Case I- Here the endurance limit is the criteria of failure. therefore the design stress (mean

stress) should be lower than the endurance limit stress in order to withstand infinite number

of cycles. So, the component are design by finding the mean value of stress.

Limited cycle fatigue for finite life design is the life between 103 to 106 cycles, for which a

line connecting points (3, 0.9 Su) and (6, Se) is drawn. From this line, life of a component

can be estimated.

Case II- When the component is designed for finite life, the S-N curve for steel is used to

design the component.

One significant limitation of the S-N curve is that the resulting plot is highly

dependent on the test conditions (e.g. the stress ratio Smin/Smax, sample geometry, sample

surface condition, and material). Using an S-N curve to predict real-world life when

conditions do not match the test conditions under which the curve was developed is dubious

at best. This severely limits the use of S-N curves in product design. On the other hand, the

ease of construction makes the S-N curve a simple and valuable tool in making relative

comparisons between materials or process variations.


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9. SODERBERG AND GOODMAN LINES

When a component is subjected to fluctuating stresses there is

mean stress (σm) as well as stress amplitude (σa) also. It is observed that the mean stresscomponent has an effect on fatigue failure when it is present in combination with an

alternating component. The fatigue diagram is shown in the figure below. Here the mean

stress is plotted the abscissa. The stress amplitude is plotted in the ordinates. The magnitudes

of the σm & σa depend on the magnitudes of the maximum and minimum forces acting onthe components. When stress amplitude is zero, the load is purely static and the criterion for

failure is Sut and Syt . These limits are plotted on the abscissa. When the mean stress is zero

the stress is completely reversing and the criterion for failure is the endurance limit S e that is

plotted on the ordinates. When the component is subjected to both components of stress σm

& σa , the actual failure occurs at different scattering points shown in the figures. There exist

a border, which divides safe region from unsafe region for various combinations of σm & σa .Different criterions are proposed to construct the borderline dividing safe zone and failure

zone. They include GERBER LINE, SODERBERG LINE AND GOODMAN LINE.


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GERBER LINE- A parabolic curve joining S e on the ordinate to S ut on the abscissa is called

gerber line.

SODERBERG LINE- A straight line joining S e on the ordinate and S yt on the abscissa is

called the soderberg line. GOODMAN LINE- A straight line joining S e on the ordinate and S ut on the abscissa is

called goodman line.

Fig- soderberg, goodman line.

The Gerber parabola fits the failure points of test data in the best

possible way. The Goodman line fits beneath the scatter of this data. Both Gerber and

Goodman line intersect at Se on the ordinate to Sut on the abscissa. However, the Goodman

line is more safe from design considerations because it is completely inside the Gerber

parabola and inside the failure points. The Soderberg line is a more conservative failure

criterion and there is no need to consider even yielding in this case. A yield line is

constructed connecting Syt on both the axes. It is called the limit of first cycle of stress. Thisis because if a part yields, it has failed, regardless of its safety in the fatigue.

We will apply following form for the equation of a straight line,

+

=

Where a and b are the intercepts of the line on the X and Y axes respectively. Applying the

above formula, the equation of the Soderberg line is given by,

+

=


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Similarly the Goodman equation will be-

+

=

The Goodman line is widely used as the criterion of fatigue failurewhen the component is subjected to mean stress amplitude. It is because of the following

reasons-

(i) the Goodman line is safe from design consideration because it is completely inside the

failure points of test data.

(ii) The equation of a straight line is simple compared to the equation of a parabolic

curve.

(iii) It is not necessary to construct a scale diagram and a rough sketch is enough to

construct fatigue diagram.

MODIFIED GOODMAN DIAGRAM:-

The components which are subjected to fluctuating stress are

designed by constructing the Modified Goodman diagram. For the purpose of design, the

problems are classified into two groups-

(i)

Components subjected to fluctuating axial or bending stresses.

(ii)

Components subjected to fluctuating torsion shear stresses.

Components subjected to fluctuating axial or bending stresses-

Here the Goodman line is modified by combining the fatigue

failure with failure by yielding. In the diagram below given, the yield strength Syt is plotted

on both the axes, and a yield line CD inclined at 450 is constructed to join these two points to

define failure by yielding. Similarly the line AF is joined Se on the ordinate with Sut on the

abscissa, which is the Goodman line. Both the lines intersect at point B. The area OABC is

called the region of safety for the components subjected to the fluctuating stresses. The

region OABC is called Modified Goodman diagram. All the points inside the Modified

Goodman diagram should cause neither fatigue nor yielding. The Modified Goodman


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diagram combines fatigue criteria as represented by the Goodman line and yield criteria as

represented by the yield line.

If the mean component of stress is very large and the alternating

component is very small, their combination will define a point in the region BCF that would

be safely within the Goodman line but would yield on the first cycle, this will result in

failure. This is the reason to modify the Goodman line.

While solving problems, a line OE with a slope of tanθ is constructed in such a way that,

=

, ,

=

=

The magnitudes of Pa and P b can be determined from max and min forces acting on the

component. Similarly it can be proved that,

=()

()

The magnitudes of the ratios can be determined from max and min bending moment actingon the component.

The point of intersection of lines AB and OE is X. The points X

indicates the dividing lines between the safe region of failure. The coordinates of the points X

represent the limiting values of stresses, which are used to calculate the dimensions of the

component. The permissible stresses are as follows-

=

, =


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Components subjected to fluctuating torsion shear stresses-

The modified Goodman diagram for fluctuating torsional shear stress is

shown below. In this diagram, the torsional mean stress is plotted on the abscissa while the

stress amplitude on the ordinates. The torsional yield stress is plotted on abscissa and the

yield line on the ordinates., which is inclined at 450 to a certain abscissa. It is interesting that

up to a certain point, the torsional mean stress has no effect on the torsional endurance limit.

Therefore, a line is drawn through Sse on the ordinate and parallel to the abscissa. The point

of intersection of this line and the yield line is B. The area OABC represents the region of

safety. A fatigue failure can be indicated if,

= ; ℎ , = + =

The permissible stress are as follows:-

= ; =

10. GERBER LINE

The Soderberg and Goodman lines are the straight lines. The

theories using such straight lines for predicting fatigue failure are called linear theories. There

are some theories that use parabolic or elliptical curves instead of straight lines. These

theories are called non linear theories. One of the most popular non-linear theories is the

Gerber theory that is based on parabolic curve. The Gerber curve is shown in the figure

below.


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The equation for the Gerber curve is as follows:-

+

= 1

Theories based on the Soderberg line or the Goodman line, as

failure criteria are conservative theories. This results in increased dimensions of the

component. The Gerber curve takes the mean through failure points. It is therefore more

accurate in predicting fatigue failure.

11. CONCLUSION

In the above report we have discussed about various conditions under

which the design against fluctuating stress can be done. The main focous before our design

will be the fatigue tests. We should keep consideration about various notch sensitive points as

well. Thus we have expalined various conditions under which we can design a component

under cyclic loads. We have also seen some of the formulas for the design consideration. We

have to take into account the Modified Goodman equations also.

If we consider this factors during our design a component for cyclic stress

then we can design in a better and more economic way.


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12. REFERENCES

1. V. B. Bhandari "Design of Machine elements" ;by McGrew hill education.

2.

Dr. P. C. Sharma & Dr. D. K. Aggarwal "Machine Design (SI units), 11th edition

reprint 2011

3. R. K Jain "Machine Design", pdf formats.

4. S. G Kulkarni "Machine Design", pdf formats.

5. https://en.m.wikipedia.org/wiki/fatigue_(material)

6. www.wikipedia.org/wiki/stress_concentration_notches

7. Shigley, J.E., C.R. Mischke, Mechanical Engineering Design, 5th Ed., McGraw-Hill,

Inc., New York, 1989.

Design for fluctuating loads

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