Working Stresses and Failure Theories

8/16/2019 Working Stresses and Failure Theories

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ME 4550 MECHANICAL

ENGINEERING DESIGN WORKING STRESSES AND FAILURETHEORIES


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AGENT OF FAILURE

Mathematical prediction of failure - most frequentlyundertaken task in mechanical design

A common misconception is that the failure of parts is onlydue to fracture

There are number of modes of failure based on other failure

mechanisms We should seek to identify failure-inducing agents and

modes of failure

With a knowledge of these; a definition can be obtained forfailure that is applicable to all possible modes of failure


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AGENT OF FAILURE

See Table on next slide Cause Agent: Force, Temperature, Reactive

chemical environment, Reactive nuclearenvironment, Reactive metallurgical

environment Level of Application: Low, Medium, or High

Time of Application: Steady, Transient orCyclic

Example: Force + High + Transient = Impact Temperature + High + Steady = Creep

Force and temperature are cause agents

High is the level of application

Transient and steady are time of application


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F AILURE INDUCING AGENT

Cause agent Level of

application

Time of

application

Force Low Steady

Temperature

Reactive chemicalenvironment

Medium Transient

Reactive nuclear

environment

Reactivemetallurgical

environment

High Cyclic


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MODES OF FAILURE

Types of Failure Modes: Elastic, Plastic,

Fracture, Material change

Duration of Failure: Sudden, Progressive

Location of Failure: Local, Surface and Volume

Failure-’change in a machine part that makes it

unable to perform its intended function’


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STRESS – STRAIN DIAGRAM

ELASTIC

REGION &

HOOKE’S

LAW

APPLIED


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STREES STRAIN CURVE OF DUCTILE

MATERIAL

Beyond yielding, the load needs to

be increased for additional strain or

elongation. This is called strain

hardening and and it is associated

with an increased resistance to slip

deformation at the microscale (for

polycrystalline materials)


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STRESS – STRAIN DIAGRAM


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DUCTILE MATERIAL – BEFORE

AND AFTER FRACTURE

Ductile material will

fracture under:

• FATIGUE

• CREEP

• IMPACT

• WORK HARDENING

• SEVERE QUENCHING

NECKING

occurs before

facture


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STRESS – STRAIN CURVE OF

BRITTLE MATERIAL

There is no necking in the fracture

of brittle material.

No NECKING

before facture


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MECHANICAL PROPERTIES OF

ENGINEERING MATERIALS TABLE 2-3 & 2-3A page 125.

FOR DUCTILEMATERIALS:

The value of

yielding in shear =

0.5 – 0.6 of the

value of yieldingin tension


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MECHANICAL FAILURE IN MATERIAL

YIELDING

Yield stress in tension = Yield stress in compression

FRACTURE – Brittle Material

Ultimate strength in compression is higher than for

tension

Fracture occurs with no yielding


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COMPRESSION TEST

Ductile Steel Brittle Cast Iron


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BENDING TEST



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TORSION TEST



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BRITTLE MATERIALS

Brittle materials do not yield, they fracture

Strength in compression >> strength in tension

Theory of failures used:

Maximum Normal Stress Theory (MNST) Modified Mohr Theory

Material strength:

Sut = Ultimate (fracture) strength in tension

Suc = Ultimate (fracture) strength in compression

Strengths are always positive numbers

Principal stresses 1, 2 and 3 can be negative or

positive.


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F AILURE THEORIES

When state of stress is uniaxial tension orcompression; limiting stress values from thetension/compression tests (which are widelypublished in material handbooks) can be used

But for complex state of stresses, predictingfailure is not straightforward

Several theories of failures have been putforward to predict failure for complex engineeringstresses

Unfortunately, no single failure theory canadequately predict failure in all combination ofengineering stresses


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F AILURE THEORIES

Failure theories have been formulated in terms of

three principal normal stresses (S1,S2, S3) at a

point

For any given complex state of stress ,

we can always

find its equivalent principal normal stresses (S1,

S2, S3)

Thus the failure theories in terms of principalnormal stresses can predict the failure due to any

given state of stress


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FAILURE THEORIES

STATIC LOADING FAILURE DUCTILE MATERIAL uses:

Max Normal Stress Theory

Max Shear Stress Theory

Max Strain Energy TheoryMax Distortion Energy Theory

BRITTLE MATERIAL uses:Max Normal Stress Theory

Coulomb-Mohr Theory

Modified Mohr-Coulomb Theory

FATIGUE LOADING FAILURE uses: SODERBERG Equation

Modified GOODMAN Equation


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STATIC DESIGN COORDINATE SYSTEM

• To predict that a design is

safe under static conditions.

• Horizontal axis is the

maximum principal stress

(σ1) and the vertical axis is

the minimum principal stress

(σ2).

• For ductile materials, the

yield strength (Sy ) in

tension and in compression

are relatively equal in

magnitude.

• For brittle materials the

ultimate compressive

strength (Suc ) is significantly

greater in magnitude than

the ultimate tensile strength

(Sut ) .


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• The four quadrants of this coordinate

system, labeled I, II, III, and IV as

shown, represent the possible

combinations of the principal stresses

(σ1, σ2) .

• As it is usually assumed that the

maximum principal stress (σ1) isalways greater than or at least equal

to the minimum principal stress (σ2) ,

combinations in the second (II)

quadrant where (σ1) would be

negative and (σ2) would be

posit ive, are not possible.• Primarily, the most common combinations are in the first (I) quadrant

where (σ1) and (σ 2) are both positive and in the fourth (IV) quadrant

where (σ 1) is positive and (σ2) is negative.

• Combinations can occur in the third (III) quadrant where (σ1) is negative,

however (σ2) must be equally or more negative.


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BRITTLE MATERIALSTHEORY OF FAILURE


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MAXIMUM NORMAL STRESS

THEORY (RANKINE’S THEORY) Failure is predicted to

occur in the multi-axialstate of stress when themaximum principalnormal stress becomesequal to or exceeds themaximum normal stressat the time of failure ina simple uni-axial stresstest using a specimen ofthe same material.

FS

T YP

FS

C YP

FS

T YP

FS

C YP

FS

T YP

FS

C YP

N S S

N S

N

S S

N

S

N

S S

N

S

3

2

1

1

2

3

U C U T

FS FS

U C U T

FS FS

U C U T

FS FS

S N N

S N N

S N N

For Brittle Material:

THESE ARE

FOR DUCTILE

MATERIAL


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THEORY (RANKINE’S THEORY) THIS THEORY IS NOT SAFE FOR DUCTILEMATERIAL

Cast iron -> Non-linear stress

strain relationship


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THEORY (RANKINE’S THEORY) THIS THEORY IS NOT SAFE FOR DUCTILEMATERIAL

Cast iron -> Non-linear stress

strain relationship

• Any combination of the principal

stresses (σ1) and (σ2) that are

inside the square is a safe design

and any combination outside the

square is unsafe. Remember, the

strengths (Sut ) and (Suc ) are

positive values.

• The mathematical expressions

representing a safe design

according to the maximum-normal-

stress theory are given,• σ1 < Sut or σ2 > − Suc

The factor-of-safety (n) for this theory

is given,

1 21 1

or

ut FS uc FS S N S N


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ULOMB-MOHR THEORY –

ITTLE MATERIAL

Any combination of the principal

stresses (σ1) and (σ2) that

are inside this enclosed area is a

safe design and any combination

outside this area is UNSAFE.

The mathematical expressions

representing a safe design

according to the Coulomb-Mohr

theory are given,

1 2

2 1

1 specifies the line in IV quad

or 1 specifies the line in II q

ut uc

ut uc

S S

S S

The factor-of-safety (n) for this theory is given,

1 2 2 11 1

or

ut uc FS ut uc FS S S N S S N

Intercept form equation of line:

1 x y

a b


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MODIFIED MOHR-COULOMB

THEORY – BRITTLE MATERIAL

The mathematical expressionsrepresenting a safe design

according to the Modified Mohr-

Coulomb theory are given,

1 2

2 1

1 1 specifies the line in IV quadrant

connecting the points (0, ) and ( , )

or 1 1 specifies the line in II

ut

ut uc uc

uc ut u t

ut

ut uc uc

S

S S S

S S S

S

S S S

quadrant

connecting the points ( ,0 ) and ( , )uc ut ut

S S S

The factor-of-safety (n) for this

theory is given,

1 2 2 11 1

1 or 1ut ut

ut uc uc FS ut uc uc FS

S S

S S S N S S S N


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1 2

2 1

1 1 specifies the line in IV quadrant

connecting the points (0, ) and ( , )

or 1 1 specifies the line in II

ut

ut uc uc

uc ut ut

ut

ut uc uc

S

S S S

S S S

S

S S S

quadrant

connecting the points ( ,0 ) and ( , )uc ut ut

S S S

1 2

2 1

1 specifies the line in IV quadrant

or 1 specifies the line in II quadrant

ut uc

ut uc

S S

S S


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COMPARISON TO EXPERIMENTAL

DATA – BRITTLE MATERIAL

• Data shown are primarily in

the first (I) and fourth (IV)

quadrants; none in the

second (II) and third (III)

quadrants.

• This is not unexpected as

combinations in the second(II) quadrant are impossible if

the principal stress (σ1) is

noted as the greater of the

two principal stresses.

• Also, combinations in the

third (III) quadrant requirethat the principal stress (σ2)

be at least equally or more

negative than the principal

stress (σ1) .


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RECOMMENDATIONS FOR BRITTLE

MATERIALFirst (I): (σ1 > 0 and σ2 > 0)• Maximum-normal-stress theory is the

most accurate. Coulomb-Mohr theory

does not apply. Modified Coulomb-

Mohr theory does not apply.

Fourth (IV): (σ1 > 0 and 0 >σ2 > − Sut )• Maximum-normal-stress theory is the

most accurate. Coulomb-Mohr theoryis okay, but conservative. Modified

Coulomb-Mohr theory does not apply.

Fourth (IV): (σ1 > 0 and

− Sut >σ2 > − Suc)• Modified Coulomb-Mohr theory is the

most accurate. Coulomb-Mohr theoryis okay, but conservative. Maximum-

normal-stress theory does not apply.


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DUCTILE MATERIALSTHEORY OF FAILURE


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• The four quadrants of this coordinate

system, labeled I, II, III, and IV as

shown, represent the possible

combinations of the principal stresses

(σ1, σ2) .

• As it is usually assumed that the

maximum principal stress (σ1) isalways greater than or at least equal

to the minimum principal stress (σ2) ,

combinations in the second (II)

quadrant where (σ1) would be

negative and (σ2) would be positive,

are not possible.

• Primarily, the most common combinations are in the first (I) quadrant

where (σ1) and (σ 2) are both positive and in the fourth (IV) quadrant

where (σ 1) is positive and (σ2) is negative.

• Combinations can occur in the third (III) quadrant where (σ1) is negative,

however (σ2) must be equally or more negative.


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MAXIMUM SHEAR STRESS THEORY

Applied to the design of ductile material

Conservative theory

Failure is predicted to occur in the multi-

axial state of stress when the maximumshearing stress magnitude becomes equal

to or exceeds the maximum shearing

stress magnitude at the time of failure in

a simple uni-axial stress test using a

specimen of the same material.


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MAXIMUM – NORMAL STRESS

THEORY – DUCTILE MATERIALS

• Any combination of the

principal stresses (σ1,

σ2) that falls inside this

square represents a safe

design, and any

combination that fallsoutside the square is

unsafe.

• The mathematical

expressions representing

a safe design according

to the maximum-normal-stress theory are given

by,

• σ1 < Sy or σ2 > − Sy

The factor-of-safety for this theory is

given,

1 21 1

or

y FS y FS S N S N


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and 0; 0 x y xy yx

P

A

max

2

ypS

max with Factor of Safety2

yp

fs fs

S

N N

UNI-AXIAL TENSION TEST

At Yield Point: x ypS

Max Shear Stress Theory

with Nfs = Factor of Safety


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MAXIMUM SHEAR STRESS

THEORY – 3D MOHR’S CIRCLE1 2

2 3

1 3

2 2

2 2

2 2

yp

fs

yp

fs

yp

fs

S S S

N

S S S

N

S S S

N

1 2

2 3

1 3

yp yp

fs fs

yp yp

fs fs

yp yp

fs fs

S S S S

N N

S S S S

N N

S S S S

N N

TRIAXIAL

STATE OF

STRESS


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MAXIMUM SHEAR STRESS

THEORY – 2D MOHR’S CIRCLE

1 2 yp

fs

S

S S N

BIAXIAL STATE

OF STRESS


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– DUCTILE MATERIALS

The maximum-shear-stress theory, given

mathematically,

1 2

1 2

2 2

y

y

S S

where the straight lines at45, one in the fourth (IV)

quadrant and one only

allowed mathematically in

the second (II) quadrant,

represents this theory

graphically.

The factor-of-safety is given

by,

1 21

y f sS N


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MAXIMUM STRAIN ENERGY

THEORY – BETRAMI THEORY

Failure is predicted to occur in the multiaxial

state of stress when the total strain energy per

unit volume becomes equal or exceeds the total

strain energy per unit volume at the time of

failure in a simple uniaxial stress test usingspecimen of the same material

UTOTAL Uuniaxial tensile test at yield ---- Failed


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STRAIN ENERGY – UNI A XIAL STRESS TEST

1

1

2 D x x

U

x

x

E

2

1

1

2

x

DU

E

2

1

1

2

yp

D

S U

E

Strain Energy:

Hooke’s Law:

At yield point: x ypS

Total Strain Energy

per a unit volume at

failure

With a factor of safety Nfs2

1

1

2

yp

D

fs

S U

E N


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STRAIN ENERGY – TRI STATE OF

STRESS

1 1 1

2 2 2 3-D 1 1 2 2 3 3

3 3 3

1

2

1 1 U

2 2

1

2

U S

U S S S S

U S

1S

2S

3S

1 1 2 3

2 2 1 3

3 3 1 2

STRESS-STRAIN RELATIONSHIP:

1 & is Poisson's Ratio

1

1

S S S E

S S S E

S S S E

2 2 2TOTAL 1 2 3 1 2 2 3 1 31

22

U S S S S S S S S S E

Total strain energy per unit volume:

Failing will not occur if: UTOTAL UUNIAXIAL


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MAXIMUM STRAIN ENERGY

THEORY

2

2 2 2

1 2 3 1 2 2 3 1 3

2

2 2 21 2 3 1 2 2 3 1 3

1 12

2 2

2

yp

fs

yp

fs

S S S S S S S S S S

E E N

S S S S S S S S S S

N

Failing will not occur if: UTOTAL UUNIAXIAL


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HYDROSTATIC LOADING

Materials that are hydrostatically loaded willhave the stresses uniform in all directions.

Very large amount of strain energy can be storedin materials without failure if they arehydrostatically loaded.

Many experiments have shown that materialscan be hydrostatically stressed to levels wellbeyond their ultimate strengths in compressionwithout failure, as this just reduces the volume ofthe specimen without changing its shape.

No distortion in the part – there is no shearstress. There is no distortion and no failure.

Thus, it appears that distortion is the cause intensile failure as well.

MAXIMUM DISTORTION ENERGY


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MAXIMUM DISTORTION ENERGY

THEORY OR VON MISES – HENCKY

THEORY

TOTAL STRAIN

ENERGY OF

LOADED PART

DUE TO HYDROSTATIC

LOADING – CHANGE IN

VOLUME

DUE TO DISTORTION –

CHANGE IN SHAPE

UT = UV + UD

+=

Normal Stresses Hydrostatic Stresses –

change in volume

Distortion Stresses –

change in shape

1S

3S

2S

V S

V S V

S

3 3V S S S

2 2V S S S

1 1V S S S


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HYDROSTATIC LOADING &

DISTORTION OF ENERGY

-

Replace , , with

-

T

v

V


S S S S S S S S S S


2 2 2

1 2 3 1 2 2 3 1 3

1 2 3

1 2 3 1 2 3

2 2 2

1 2 3 1 2 2 3 1 3

12

2

3

1 2

26

-

-

D T V

D

U U U

S S S S S S S S S E

S S S S S S S S S E


2 2 21 2 3 1 2 2 3 1 3

2 2 2

1 2 3 1 2 2 3 1 3

2 2 2

1 2 3 1 2 2 3 1 3

12

2

1 22

6

1

3

STRAIN ENERGY DUE TO HYDROSTATIC LOADING - UV:

STRAIN ENERGY DUE TO DISTORTION - UD:


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DISTORTION OF ENERGY OF UNIAXIAL

TEST AT YIELD POINT

yp fs

NORMAL STRESS:

and

At yield point the stress is S and if N Safety Factor,

x

yp

fs

PS S S

A

S S

N

1 2 3

1

0

-UNIAXIAL

-UNIAXIAL

D

yp

D

fs

U S E

S U

E N

2

1

2

1

3

1

3

DU S S S S S S S S S

E

2 2 2

1 2 3 1 2 2 3 1 3

1

3

P

P

UD for Uniaxial state of stress:


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DISTORTION ENERGY THEORY OR VON MISES – HENCKY THEORY

Failure is predicted to occur in the multiaxial

state of stress when the distortion energy per

unit volume becomes equal or exceeds the

distortion energy per unit volume at the time of

failure in a simple uniaxial stress test usingspecimen of the same material

UD-TOTAL UD-uniaxial tensile test at yield ---- Failed


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MAXIMUM DISTORTION ENERGY THEORY

OR VON MISES – HENCKY THEORY

-UNIAXIAL

D

yp

D

fs


S U

E N

2 2 2

1 2 3 1 2 2 3 1 3

2

1

3

1

3

or yp

fs

yp

fs

S S S S S S S S S S

E E N

S S S S S S S S S S

N

2

2 2 2

1 2 3 1 2 2 3 1 3

2

2 2 2

1 2 3 1 2 2 3 1 3

1 1

3 3

yp

fs

yp

fs

S S S S S S S

N

S S S S S S S

N

22 2 2

1 2 2 3 3 1

2 2 2

1 2 2 3 3 1

2

2

VM

S S S S S S

2 2 2

1 2 2 3 3 1

2

FAILURE WILL NOT OCCUR IF UD-TOTAL UD-UNIAXIAL

Alternately, Von Mises Stress:


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DISTORTION ENERGY THEORY –

DUCTILE MATERIALS

For a safe design accordingto the distortion-energy

theory,

The above expression in

represents the equation of

an ellipse inclined at 45

as shown in the Figure.

This ellipse passes

through the six corners of

the enclosed shape.

2 2 2

1 2 1 2 yS

2 2

1 2 1 2 1

y fsS N

FACTOR OF SAFETY:


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COMPARISON TO EXPERIMENTAL

DATA – DUCTILE MATERIALS

Note that there is noexperimental data in the

second (II) and third (III)

quadrants. This is not

unexpected as combinations

in the second (II) quadrant are

impossible if the maximumprincipal stress (σ1) is greater

than or at least equal to the

minimum principal stress (σ2) .

Also, combinations in the third

(III) quadrant require that the

principal stress (σ2) be at least

equally or more negative than

the principal stress (σ1) .


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RECOMMENDATION FOR DUCTILE

MATERIALSFirst (I): Distortion-energy

theory is the most

accurate. Maximum-

normal-stress theory

is okay, but conservative.

Maximum-shear-stress

theory does not apply.

Fourth (IV): Distortion-

energy theory is the most

accurate. Maximum-

shear-stress theory is

okay, but conservative.Maximum-normal-stress

theory does not apply.


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STRESS CONCENTRATION

Stress concentration is caused by the suddenchanges in geometry.

Sudden change in geometry: Hole

Shoulder fillet

Groove

Notch

TWO STRESS

CONCENTRATION

FACTORS

K or Kt – STRESS

CONCENTRATION

FACTOR for STATIC

LOADING

Kf – STRESS

CONCENTRATION

FACTOR for CYCLIC

LOADING


53/108


The theoretical stress concentration is defined by:

max maxMax Stress at the section of interest

Nominal Stress at the section of interestt o nom

K

ots

K

max

-> Stress-concentration factor in SHEAR

Stress-concentration factors are dependent on the geometry of the

machine element, not on the material used. However, some materials

are more sensitive to stress concentrations, or notches, so the stress-

concentration factors will be modified according to their notch

sensitivity.


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K OR K T – GEOMETRIC STRESS

CONCENTRATION FACTOR

max

Nominal Stress with the hole

0

03

Stress distribution for semi-infinite plate with a hole. Plate in tension:


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KT

- PLATE WITH HOLE

( )nom

P P

A W D h

max

is geometric stress concentration factor

t nom

t

K

K

Nominal tensile stress:

Max tensile stress in the vicinity of the hole:


56/108

PLATE WITH NOTCH UNDER

TENSION

nom

P P

A d h

max

is geometric stress concentration factor

t nom

t

K

K

d = D – 2r

r is the radius of notch

Nominal tensile stress: Max tensile stress in the vicinity of the hole:


57/108

LOADING WITHOUT AND WITH



58/108

FILLETS ON SHAFTS


59/108

FORCE-FLOW ANALOGY

The forces (stresses)

flow around the

contours is similar tothe flow of

incompressible fluid

inside a step duct.

The force-flow analogyfor contoured parts

Analogy

DESIGN MODIFICATIONS TO REDUCE


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STRESS CONCENTRAIONS AT A SHARP

CORNER


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STRESS CONCENTRATION FACTOR

FOR CYCLIC LOADING - K F

Materials are not homogenous or free from defects,therefore it is necessary to define fatigueconcentration factor K f :

specimennotched aof limitEndurancenotchesof freespecimenaof limitEndurance f K

• In cyclic loading, the effect of the notch (hole, groove,

fillet, an abrupt change in cross section, or any

disruption to the smooth contours of a part) isusually less predicted. The term notch sensitivity q

has been applied to this behavior.


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NOTCH SENSITIVITY Q

Due to existence of irregularities ordiscontinuities (holes, grooves or notches)

in part, high stress will occur in the

immediate vicinity of the discontinuity.Some materials are not fully sensitive to

the presence of notches, for these a

reduced value of K t can be used.

max

is fatigue stress-concentration factor

f o

f

K

K


63/108

NOTCH SENSITIVITY

Notch sensitivity q may be defined as thedegree to which the theoretical effect of stress

concentration is actually reached.

t

0

is fatigue stress concentration factor

or is geometric stress concentration factor

f

f

f

K q qK

K q K

K

K K

1 11

1 1

Eqns. (8) and (9), page 147

Use of q = 1 ----→ The design will be on

safe side.

fs

shear

s

K q

K

1

1

q

STRESS CONCENTRATION FACTOR FOR


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STRESS CONCENTRATION FACTOR FOR

CYCLIC LOADING – K F AND NOTCH

SENSITIVITY Q

K f is a stress-concentration factor reducedfrom Kt because of lessened sensitivity to

notches.

f t K q K 1 1

If 0 1 f

q K The material has NO

sensitivity to notches.

Cast Iron is insensitive to

notches K f 1

If 1 f t

q K K The material has FULL

sensitivity to notchesIf a value of the notch

sensitivity is not known, use

a value of 1 to be safe.


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INDEX OF SENSITIVITY OR NOTCH

SENSITIVITY

Materials have different sensitivity to stress concentrations, whichis referred to as the notch sensitivity of material.

q for cast irons is

varying from 0 to

about 0.2 (very low)

q

NOTCH SENSITIVITY FOR STEEL & AL –


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REVERSED BENDING OR REVERSED AXIAL

LOADS


67/108

NOTCH SENSITIVITY FOR MATERIAL

IN REVERSED TORSION


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STATIC VS FATIGUE FAILURES

STATIC FAILURE:

They usually developed a very large deflection,

because the stress has exceeded the yield

strength.

Thus static failures give visible warning inadvance

The part is replaced before fracture occurs

FATIGUE FAILURE:

Fatigue failure give NO warning!

It is sudden and total, and hence dangerous!


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FATIGUE FAILURE THEORIES

Many machine parts are subjected to a loading cycles in which the

stress is not steady but continuously varying. Failures in machine

parts are generally caused by such repeated loadings and at stress

that are considerably below the yield point. This is called fatigue

failure and it resembles the failure of brittle material.


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CYCLIC LOADING OF FLAT SPRING


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FATIGUE-FAILURE MODELS


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FATIGUE REGIMES

Based on the number of stress or strain cycles thepart is expected to undergo in its lifetime, we can

classify two regimes: LCF – LOW CYCLE FATIGUE

HCF – HIGH CYCLE FATIGUE


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STRESS-LIFE APPROACH

The oldest and easiest of the three models to use. Most often used for High-Cycle Fatigue (HCF) where

part is expected to undergo more than about 103

cycles of stress.

Work best when the load amplitudes are predictableand consistent over the life of the part.

It is a stress-based model, which determines a fatigue

strength and/or an endurance limit for material so

that the cyclic stresses can be kept below that leveland failure avoided for the required number of cycles


74/108

STRESS-LIFE APPROACH

Part is designed based on the material’s fatiguestrength or endurance limit and factor of safety.

Assume that stresses and strains everywhere remain

in elastic zone.

No local yielding occurs to initiate a crack. Least accurate in defining stress/strain states in the

part, especially for LCF finite-life where N is less

than 103 cycles and stress is high enough to cause a

local yielding. For certain materials, stress-life approach allow the

design of parts for infinite life under cyclic loading.


75/108

STRAIN-LIFE APPROACH

Strain-based model gives a reasonablyaccurate picture of crack initiationstage.

Good for designing part under fatigue

loading and high temperature combined – creep effects can be included.

Most often used for LCF finite-lifeproblems where the cyclic stresses are

high enough to cause local yielding.Most complicated of three models to use

and requires a computer solution


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LEFM APPROACH

Best model for the crack propagation stage – dueto fracture mechanics theory.

Apply to LCF finite-life problem where the cyclicstresses are known to be high enough to cause

the formation of cracks. Most useful in predicting the remaining life of

cracked parts in service.

Most often used in conjunction withnondestructive testing (NDT) in a periodic

service-inspection program, especially in theaircraft/aerospace industry.

More accurate results when a detectable andmeasurable crack already exists.

MACHINE DESIGN


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MACHINE-DESIGN

CONSIDERATIONS

STRESS-LIFE (S-N) approach:

Suitable for large class of rotating machinery

because the required lives are usually in the

HCF range:

Automobile-engine crankshaft – the crankshaft and

most other rotating and oscillating components in the

engine will see about 2.5E8 cycles in 100,000 miles

Automated production machinery in industry such as

filling soft-drink cans etc.

MACHINE DESIGN


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MACHINE-DESIGN

CONSIDERATIONS

STRAIN-LIFE (ε-N) approach: Suitable for transportation (service) machinery

because the required lives are usually in the LCF

range:

Airframe of an airplane, the hull of a ship, the chassis of a

land vehicle.

For aircraft/ship -> time-load history can be quite

variable due to storms, gusts/waves, hard

landings/dockings etc.

For land vehicle – also time-load history is variable due

to overloads, potholes etc. The chance of higher-than-design loads causing local

yielding is always present and consequently crack growth

exists.

MACHINE DESIGN


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MACHINE-DESIGN

CONSIDERATIONS

LEFM and STRAIN-LIFE (ε-N) approach: The LEFM or strain-life models (or both) use

simulated and experimental load-time histories for

more accurately predict failure.

Example of the use of ε-N and LEFM models: Design and analysis of gas turbine rotor blades subjected to

high stresses at high temperatures and go though LCF

thermal cycles at start up and shut down.

TYPICAL S N (STRESS LIFE)


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TYPICAL S-N (STRESS-LIFE)

DIAGRAM FOR STEELFatigue failures due to:

• Bending – most common• Torsion – next

• Axial loading – rare.

Parts that are subject to variable

loading will lose strength with

time and may fail after a certainnumber of cycles.

Endurance limit

stress

We assume that stresses below

endurance limit will NOT cause

failure regardless of the number of

repetitions.

Cyclic stress

The most important thing to observe in an S-N diagram, if the

material being tested is ferrous like steel, is that the straight line atthe lower right of the diagram becomes horizontal somewhere

between (N = 106) cycles and (N = 107) cycles and stays horizontal

thereafter .


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FATIGUE FAILURE THEORIES

TWO FATIGUE FAILURE

THEORIES

SOLDERBERG

CRITERION

MODIFIED GOODMAN

CRITERION

The fatigue or endurance limit stress Se is defined as the

maximum value of the completely reversed bending

stress, which a plain specimen can sustain for 10 millions

or more load cycles without failure. It can be assumed that

it will last indefinitely.


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ROTATING MACHINE LOADING

THREE ROTATING

MACHINE LOADINGS

FULLY REVERSED FLUCTUATINGREPEATED

PARAMETERS FOR ROTATING


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PARAMETERS FOR ROTATING

MACHINE LOADING

max minS

max min

r aS

2

max min

avg mS

2

max

min

R

m

a A

The stress range S or :

The alternating component Sr or a:

The mean component Savg or m:

Stress Ratio: Amplitude Ratio:


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SOLDERBERG EQUATION

Solderberg equation:

max min

max min

where:

( )

yp yp

avg r f

e fs

avg

r

f t

S S S S K

S N

S S S

S S S

K q K

2

2

1 1

ypS yp

fs

S

N avgS

f r K S

e

fs

S

N

eS

Eqn. (10), page 157


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S-N DIAGRAM

LCF HCF

UNCORRECTED ENDURANCE LIMIT


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UNCORRECTED ENDURANCE LIMIT

– S’ E


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CORRECTED ENDURANCE LIMIT - SE

ENDURANCE LIMIT


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ENDURANCE LIMIT

Curve C is typical for most nonferrous materials (aluminum) which do

not exhibit an endurance limit. For such materials, the number of cycles

to failure should be reported for the given endurance strength.

Unfortunately, for

nonferrous materials

like aluminum there

is no endurance limit,meaning the test

specimen will

eventually fail atsome number of

cycles, usually near

(N = 108) cycles, nomatter how much the

stress level is

reduced. This is why

critical aluminumparts, especially

those in aircraft

where the number ofreversed loadings

can become very

high in a short period

of time, must beinspected regularly

and replaced prior toreaching an unsafenumber of cycles.


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MODIFIED GOODMAN EQUATION

TWO EQUATIONS OF MODIFIED


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TWO EQUATIONS OF MODIFIED

GOODMAN THEORY

max min

max min

where:

( )

u uavg r f

e fs

avg

r

f t

S S S S K

S N

S S S

S S S

K q K

2

2

1 1 yp

avg r f

fs

S S S K

N

First equation:

Replace Syp in Solderberg

equation with Su

Second equation:

Both equations must be satisfied, if one fails, we have unsafe conditions.

Eqns. (11a) and (11b), page 157

VARIOUS FAILURE LINES FOR


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VARIOUS FAILURE LINES FOR

FLUCTUATING STRESSES

Gerber parabola is a good fit to experimental data, making it useful for the

analysis of failed part.

Modified Goodman line is a more conservative and commonly used failure

criterion when designing parts subjected to mean plus alternating stresses.

The Solderberg line is less often used, as it is overly conservative

LINEAR HYPOTHESE


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LINEAR HYPOTHESE

Equation of straight line in

intercept form:

1 x y

a b

For Soderberg line:

1m a

yt e

S S

S S

For Goodman line:

11 or

f am a m

ut e ut e fs

K S S S S

S S S S N

For Gerber parabola:

2

1m a

ut e

S S

S S

LINEAR HYPOTHESE


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LINEAR HYPOTHESE

For Soderberg line:

1 m

a e

yt

S S S

S

For Goodman line:

For Gerber parabola:

1

1

m

a e

ut

e ma

f fs ut

S S S S

S S S

K N S

2

21

m

a e

ut

S S S

S


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MODIFIED GOODMAN DIAGRAM


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DESIGN FOR FINITE LIFE An infinitely large

number of fullyreversed stress

(40,000 psi) may

take place before

the material

fracture.

This design is called for

finite life.

From N = 1000 cycles to N = 1,000,000 cycles DESIGN FOR FINITE

LIFE


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DESIGN FOR FINITE LIFE

LOG

Scale

LOGScale

DESIGN FOR FINITE LIFE – BASQUIN’S


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DESIGN FOR FINITE LIFE – BASQUIN S

EQUATION

The equation for a straight line on log-log (logarithmic) paper is,log log log

or

where, y-intercept & slope

m

y m x b

y bx

b m

Parabolic for m = positive

or hyperbolic (m =

negative) curves when

plotted in rectangular

coordinate paper.

DESIGN FOR FINITE LIFE –


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DESIGN FOR FINITE LIFE

BASQUIN’S EQUATION

S-N curve for

avg = 0

The curve plot

is transformed

into straight-line in log-log

plot.

DESIGN FOR FINITE LIFE –


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S G OR

BASQUIN’S EQUATION

Straight line in logarithmicscale.

The stress value at 103 cycles

(starting cycle for HCF) is 0.9

ult = 0.9 x 90000 psi = 81,000

psi.

The equation for straight line:

--- Power Equations

log log log

B m

r

r

AN y bx

A B N

DESIGN FOR FINITE LIFE – BASQUIN’S


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DESIGN FOR FINITE LIFE BASQUIN S

EQUATION

B

r AN

log log .e u

e

B

B

A

6

0 9

3

10

B

r N A

1

Alternating stress r:

Number of cycles N:

The coefficients for Basquin’s equation A & B:

Eqns. (12) and (13), page 161

Eqns. (14) and (15), page 162

MINER’S EQUATION FOR CUMULATIVE


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MINER S EQUATION FOR CUMULATIVE

DAMAGE

If the machine part is to operate for a finite timeat stress levels exceeding the endurance limit,

then we must examine the cumulative damage.

Miner’s equation is used for machine part

subjected to various fully reversed stress level. Failure occurs when:

...

nn n

N N N

31 2

1 2 3

1

ENDURANCE

LIMIT

Stress levels are

higher than

endurance limit

MINER’S EQUATION FOR CUMULATIVE


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MINER S EQUATION FOR CUMULATIVE

DAMAGE

Failure occurs when: ...nn n

N N N

31 2

1 2 31

MINER’s

EQUATION

Proportionate Damage D:

31 2

1 2 3

1 2 3

1 2 3

31 2

1 2 3

; ; and so on.

1

... 1

i

i

i

n nn n D D D D

N N N N

D D D

nn n

N N N

Proportion of Total Life N:

1 1 2 2

Actual life of the part

; and so on.

N

n N n N

31 2

1 2 3

31 2

1 2 3

From Miner's equation,

1

1

N N N

N N N

N N N N

1 2 3

1 2 3

Note that

1

N N N N

FATIGUE LIFE PREDICTION WITH RANDOMLY

VARYING COMPLETELY REVERSED STRESSES


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VARYING, COMPLETELY REVERSED STRESSES

Stresses (including stress concentration factor Kf ) at the critical notch of a part

fluctuated randomly as indicated in Figure below. The stresses could bebending, torsional, or axial – or even equivalent bending stresses resulting

from general biaxial loading. The plot shown represents what is believed to be

20 seconds of operation. The material is steel, and the appropriate S-N curve

is given in the Figure shown. This curve is corrected for load, gradient, and

surface. Estimate the fatigue life of the part.



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Endurance limit = 60,000 psi.

There are 8 stress cycles above the

endurance limit of 60 Ksi:

Five at 80 Ksi → n1 = 5 cycles

Two at 90 Ksi → n2 = 2 cycles One at 100 Ksi → n3 = 1 cycle



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There are 8 stress cycles above theendurance limit of 60 Ksi:


Two at 90 Ksi → n2 = 2 cycles

One at 100 Ksi → n3 = 1 cycle

S-N curve shows:

Each 80-ksi cycle uses one part in 105 of the life → N1 = 105 cycles,

Each 90-ksi cycle uses one part in 3.8x104 of the life → N2 = 3.8 x 104 cycles,

Each 100-ksi cycle uses one part in 1.6x104 of the life → N3 = 1.6 x 104 cycles.

Ni is number of cycles-to-failure at stress

level Si

ni is number of cycles at fully reversedstress level Si




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There are 8 stress cycles above theendurance limit of 60 Ksi:


Two at 90 Ksi → n2 = 2 cycles

One at 100 Ksi → n3 = 1 cycle

S-N curve shows:

Each 80-ksi cycle uses one part in N1 = 105 of the life,

Each 90-ksi cycle uses one part in N2 = 3.8x104 of the life,

Each 100-ksi cycle uses one part in N3 = 1.6x104 of the life.

Miner’s rule:

31 2

1 2 3

5 4 4

1

5 3 10.0001651

10 3.8 10 1.6 10

nn n N N N

Failure occurs




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Miner’s rule:

31 2

1 2 3

5 4 4

1

5 3 10.0001651

10 3.8 10 1.6 10

nn n

N N N

For the fraction of life consumed to be unity, the 20-second test time must

be multiplied by

1

Fatigue Life 20 121,138.7038 seconds0.0001651

121,138.7038Fatigue Life 2018.9784 minutes

60

2018.9784Fatigue Life 33.6496 hours

60


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Working Stresses and Failure Theories

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