5192016

1

Aircraft Structural Integrity Module 2 Fatigue

Lecture 1

Todayrsquos LectureIntroduction to Fatigue Some Basics

Fatigue Crack in Boeing 7373-300 Emergency landing of Southwest flt 812 Apr 2013

5192016

2

F A T I G U E Basics

What is fatigue

Fatigue is hellipbull a process in which damage accumulates due to

repetitive loads which may be well below yieldstress or static strength

bull a fracture phenomenon occurring after a largenumber of load cycles where a single load of thesame magnitude will do no harm

bull a form of failure that occurs in structuressubjected to dynamic and fluctuating stressesUnder these circumstances it is possible forfailure to occur at a stress level considerablylower than the tensile or yield strength for astatic load

5192016

3

Stress ndash Life (S-N) Curve

Fatigue Cyclic (even if low) loads cause failures

Steel Ferrous Alloys

Many non-Ferrous Alloys

eg Al Alloys

5192016

4

Why should the cyclic loads lead to failure at lower loads

Origin of fatigue phenomenon

Fatigue Process Crack Initiation and Growth

Free Surface

STAGE I CRACK INITIATION Several micro-cracks originate and then coalesce into a macro-crack

STAGE II CRACK GROWTH

LOADING DIRECTION

Dislocations Slip Microcrack Macrocrack

5192016

5

Dislocations

Slip bands and micro-crack initiation

①

②

③

④

~ 01 μ

~ 01 μ

Persistent Slip Bands

5192016

6

Features of Fatigue Phenomenon

bull A gradually progressive process

bull Slow development of damage in the early stage followed by rapid growth towards the end

bull First stage crack initiation phase

ndashMay occupy most of the life in most cases (eg nicely machined parts small cyclic loads )

ndashUsually confined to a small area of high local stress

ndashAdjacent areas with only slightly lower stresses may not see any fatigue damage

ndashusually several independent micro-cracks grow and coalesce to form one dominant crack

hellip 1

Dislocations Slip Microcrack Macrocrack Crack Propagation Failure

Features of Fatigue Phenomenon

bull Second Stage Crack Propagation phasendashThis crack grows slowly under steady fatigue loading

but starts to accelerate as the net section decreases increasing the local stress at the crack-front

bull Third and Final Stage Failure ndashFailure occurs as an unstable fracture when the

remaining area is too small to support the load

hellip 2

bull Stages create distinctive features of the fracture surface - used to identify fatigue failure in failure investigations

5192016

7

Tell-tale marks of Fatigue - Striations

Features of Fatigue Phenomenon

bull Some materials exhibit endurance limits ie a stress below which the life is infinitendash Steels typical endurance limit - 40 -

60 of YS

ndash solutes (carbon nitrogen) present in the material pin the dislocations and prevent their motion at small strains

ndash Al alloys No endurance limits

ndash Related to the absence of dislocation-pinning solutes

hellip 3

bull At large Nf the lifetime is dominated by nucleation

ndash Therefore strengthening the surface (shot penning) is beneficial to delay crack nucleation and extend life

5192016

8

S

N

bull Typically load life and load life

bull Experimentally-determined ndash Rotating bending mc round specimens

ndash Electrohydraulic mc flat (or other) specimens

bull The practical limit of experimental testing has been 106 or 107 cycles due to impractical test times frequency limitations of machines and artefacts of higher frequencies

bull Adjustment for mean stress using Goodman Relation

Observations on S-N Curve

S

N

bull X-Axis No of Cycles On Log10 scale Log10 N

bull Y-Axis Stress ndash Either Stress Amplitude (Alternating stress) or Max

Stress

ndash Alternating Stress (ie Stress Amplitude) is the most dominant aspect of load that influences fatigue

ndash Either linear scale or Log10 scale

bull Curve is obtained for a given stress ratio R

bull R = (Min Stress Max Stress)

Observations on S-N Curve

5192016

9

S

N

bull S-N curve data is generally developed under Constant Amplitude ndash we will discuss later variable amplitude

bull The curve is dependent on many conditions ndash Loading ratio min load max load (R-ratio)

ndash Material

ndash stress concentrators

ndash surface finish

ndash frequency cyclic shape dwell etc hellip Not much effect unless in corrosive environment

Observations on S-N Curve

S

N

High cycle fatigue vs Low cycle fatigue

bull HCF Life usually more than 104 cycles ndash The distinction between high cycle fatigue and low

cycle fatigue is reflected in most design standards as high cycle fatigue design standards normally give S-N curves starting with N = 104 cycles

bull LCF Life usually less than 104 cycles ndash For low cycle fatigue design the stress range

concept is not immediately valid since typically the low cycle fatigue strength is governed by (large inelastic) strains

Arbitrary Could be 103

Arbitrary Could be 103

5192016

10

HCF and LCF

bull HCFndash Typical of rotating components machineriesndash 100 RPM = 6000 cycles per hr = 144x105 day or

taking 8 hrday operation5x107 yrndash Automobile crankshaft may accumulate 108 cycles in

100000 km runndash Reciprocating machineries springshellip etc etc

bull LCFndash Airplane Ground-Air-Ground Cyclesndash Shipsndash Vehicle Chassishellip

Various types of fatigue

bull Fretting fatiguendash Fatigue associated with or aided by repetitive rubbing

of surfaces causing wear and cracks also often accompanied by corrosion Such cracking provides sites for fatigue damage and decreases fatiguestrength of materials operating under cycling stress

bull Sonic fatigue (Acoustic fatigue) ndash Fatigue caused by acoustic vibrations acoustic

pressures engine noise can generate high stresses

bull Thermomechanical fatigue ndash Mechanical loads (cyclic) also accompanied by

thermal cycles

5192016

3

Stress ndash Life (S-N) Curve

Fatigue Cyclic (even if low) loads cause failures

Steel Ferrous Alloys

Many non-Ferrous Alloys

eg Al Alloys

5192016

4

Why should the cyclic loads lead to failure at lower loads

Origin of fatigue phenomenon

Fatigue Process Crack Initiation and Growth

Free Surface

STAGE I CRACK INITIATION Several micro-cracks originate and then coalesce into a macro-crack

STAGE II CRACK GROWTH

LOADING DIRECTION

Dislocations Slip Microcrack Macrocrack

5192016

5

Dislocations

Slip bands and micro-crack initiation

①

②

③

④

~ 01 μ

~ 01 μ

Persistent Slip Bands

5192016

6

Features of Fatigue Phenomenon

bull A gradually progressive process

bull Slow development of damage in the early stage followed by rapid growth towards the end

bull First stage crack initiation phase

ndashMay occupy most of the life in most cases (eg nicely machined parts small cyclic loads )

ndashUsually confined to a small area of high local stress

ndashAdjacent areas with only slightly lower stresses may not see any fatigue damage

ndashusually several independent micro-cracks grow and coalesce to form one dominant crack

hellip 1

Dislocations Slip Microcrack Macrocrack Crack Propagation Failure

Features of Fatigue Phenomenon

bull Second Stage Crack Propagation phasendashThis crack grows slowly under steady fatigue loading

but starts to accelerate as the net section decreases increasing the local stress at the crack-front

bull Third and Final Stage Failure ndashFailure occurs as an unstable fracture when the

remaining area is too small to support the load

hellip 2

bull Stages create distinctive features of the fracture surface - used to identify fatigue failure in failure investigations

5192016

7

Tell-tale marks of Fatigue - Striations

Features of Fatigue Phenomenon

bull Some materials exhibit endurance limits ie a stress below which the life is infinitendash Steels typical endurance limit - 40 -

60 of YS

ndash solutes (carbon nitrogen) present in the material pin the dislocations and prevent their motion at small strains

ndash Al alloys No endurance limits

ndash Related to the absence of dislocation-pinning solutes

hellip 3

bull At large Nf the lifetime is dominated by nucleation

ndash Therefore strengthening the surface (shot penning) is beneficial to delay crack nucleation and extend life

5192016

8

S

N

bull Typically load life and load life

bull Experimentally-determined ndash Rotating bending mc round specimens

ndash Electrohydraulic mc flat (or other) specimens

bull The practical limit of experimental testing has been 106 or 107 cycles due to impractical test times frequency limitations of machines and artefacts of higher frequencies

bull Adjustment for mean stress using Goodman Relation

Observations on S-N Curve

S

N

bull X-Axis No of Cycles On Log10 scale Log10 N

bull Y-Axis Stress ndash Either Stress Amplitude (Alternating stress) or Max

Stress

ndash Alternating Stress (ie Stress Amplitude) is the most dominant aspect of load that influences fatigue

ndash Either linear scale or Log10 scale

bull Curve is obtained for a given stress ratio R

bull R = (Min Stress Max Stress)

Observations on S-N Curve

5192016

9

S

N

bull S-N curve data is generally developed under Constant Amplitude ndash we will discuss later variable amplitude

bull The curve is dependent on many conditions ndash Loading ratio min load max load (R-ratio)

ndash Material

ndash stress concentrators

ndash surface finish

ndash frequency cyclic shape dwell etc hellip Not much effect unless in corrosive environment

Observations on S-N Curve

S

N

High cycle fatigue vs Low cycle fatigue

bull HCF Life usually more than 104 cycles ndash The distinction between high cycle fatigue and low

cycle fatigue is reflected in most design standards as high cycle fatigue design standards normally give S-N curves starting with N = 104 cycles

bull LCF Life usually less than 104 cycles ndash For low cycle fatigue design the stress range

concept is not immediately valid since typically the low cycle fatigue strength is governed by (large inelastic) strains

Arbitrary Could be 103

Arbitrary Could be 103

5192016

10

HCF and LCF

bull HCFndash Typical of rotating components machineriesndash 100 RPM = 6000 cycles per hr = 144x105 day or

taking 8 hrday operation5x107 yrndash Automobile crankshaft may accumulate 108 cycles in

100000 km runndash Reciprocating machineries springshellip etc etc

bull LCFndash Airplane Ground-Air-Ground Cyclesndash Shipsndash Vehicle Chassishellip

Various types of fatigue

bull Fretting fatiguendash Fatigue associated with or aided by repetitive rubbing

of surfaces causing wear and cracks also often accompanied by corrosion Such cracking provides sites for fatigue damage and decreases fatiguestrength of materials operating under cycling stress

bull Sonic fatigue (Acoustic fatigue) ndash Fatigue caused by acoustic vibrations acoustic

pressures engine noise can generate high stresses

bull Thermomechanical fatigue ndash Mechanical loads (cyclic) also accompanied by

thermal cycles

5192016

4

Why should the cyclic loads lead to failure at lower loads

Origin of fatigue phenomenon

Fatigue Process Crack Initiation and Growth

Free Surface

STAGE I CRACK INITIATION Several micro-cracks originate and then coalesce into a macro-crack

STAGE II CRACK GROWTH

LOADING DIRECTION

Dislocations Slip Microcrack Macrocrack

5192016

5

Dislocations

Slip bands and micro-crack initiation

①

②

③

④

~ 01 μ

~ 01 μ

Persistent Slip Bands

5192016

6

Features of Fatigue Phenomenon

bull A gradually progressive process

bull Slow development of damage in the early stage followed by rapid growth towards the end

bull First stage crack initiation phase

ndashMay occupy most of the life in most cases (eg nicely machined parts small cyclic loads )

ndashUsually confined to a small area of high local stress

ndashAdjacent areas with only slightly lower stresses may not see any fatigue damage

ndashusually several independent micro-cracks grow and coalesce to form one dominant crack

hellip 1

Dislocations Slip Microcrack Macrocrack Crack Propagation Failure

Features of Fatigue Phenomenon

bull Second Stage Crack Propagation phasendashThis crack grows slowly under steady fatigue loading

but starts to accelerate as the net section decreases increasing the local stress at the crack-front

bull Third and Final Stage Failure ndashFailure occurs as an unstable fracture when the

remaining area is too small to support the load

hellip 2

bull Stages create distinctive features of the fracture surface - used to identify fatigue failure in failure investigations

5192016

7

Tell-tale marks of Fatigue - Striations

Features of Fatigue Phenomenon

bull Some materials exhibit endurance limits ie a stress below which the life is infinitendash Steels typical endurance limit - 40 -

60 of YS

ndash solutes (carbon nitrogen) present in the material pin the dislocations and prevent their motion at small strains

ndash Al alloys No endurance limits

ndash Related to the absence of dislocation-pinning solutes

hellip 3

bull At large Nf the lifetime is dominated by nucleation

ndash Therefore strengthening the surface (shot penning) is beneficial to delay crack nucleation and extend life

5192016

8

S

N

bull Typically load life and load life

bull Experimentally-determined ndash Rotating bending mc round specimens

ndash Electrohydraulic mc flat (or other) specimens

bull The practical limit of experimental testing has been 106 or 107 cycles due to impractical test times frequency limitations of machines and artefacts of higher frequencies

bull Adjustment for mean stress using Goodman Relation

Observations on S-N Curve

S

N

bull X-Axis No of Cycles On Log10 scale Log10 N

bull Y-Axis Stress ndash Either Stress Amplitude (Alternating stress) or Max

Stress

ndash Alternating Stress (ie Stress Amplitude) is the most dominant aspect of load that influences fatigue

ndash Either linear scale or Log10 scale

bull Curve is obtained for a given stress ratio R

bull R = (Min Stress Max Stress)

Observations on S-N Curve

5192016

9

S

N

bull S-N curve data is generally developed under Constant Amplitude ndash we will discuss later variable amplitude

bull The curve is dependent on many conditions ndash Loading ratio min load max load (R-ratio)

ndash Material

ndash stress concentrators

ndash surface finish

ndash frequency cyclic shape dwell etc hellip Not much effect unless in corrosive environment

Observations on S-N Curve

S

N

High cycle fatigue vs Low cycle fatigue

bull HCF Life usually more than 104 cycles ndash The distinction between high cycle fatigue and low

cycle fatigue is reflected in most design standards as high cycle fatigue design standards normally give S-N curves starting with N = 104 cycles

bull LCF Life usually less than 104 cycles ndash For low cycle fatigue design the stress range

concept is not immediately valid since typically the low cycle fatigue strength is governed by (large inelastic) strains

Arbitrary Could be 103

Arbitrary Could be 103

5192016

10

HCF and LCF

bull HCFndash Typical of rotating components machineriesndash 100 RPM = 6000 cycles per hr = 144x105 day or

taking 8 hrday operation5x107 yrndash Automobile crankshaft may accumulate 108 cycles in

100000 km runndash Reciprocating machineries springshellip etc etc

bull LCFndash Airplane Ground-Air-Ground Cyclesndash Shipsndash Vehicle Chassishellip

Various types of fatigue

bull Fretting fatiguendash Fatigue associated with or aided by repetitive rubbing

of surfaces causing wear and cracks also often accompanied by corrosion Such cracking provides sites for fatigue damage and decreases fatiguestrength of materials operating under cycling stress

bull Sonic fatigue (Acoustic fatigue) ndash Fatigue caused by acoustic vibrations acoustic

pressures engine noise can generate high stresses

bull Thermomechanical fatigue ndash Mechanical loads (cyclic) also accompanied by

thermal cycles

5192016

5

Dislocations

Slip bands and micro-crack initiation

①

②

③

④

~ 01 μ

~ 01 μ

Persistent Slip Bands

5192016

6

Features of Fatigue Phenomenon

bull A gradually progressive process

bull Slow development of damage in the early stage followed by rapid growth towards the end

bull First stage crack initiation phase

ndashMay occupy most of the life in most cases (eg nicely machined parts small cyclic loads )

ndashUsually confined to a small area of high local stress

ndashAdjacent areas with only slightly lower stresses may not see any fatigue damage

ndashusually several independent micro-cracks grow and coalesce to form one dominant crack

hellip 1

Dislocations Slip Microcrack Macrocrack Crack Propagation Failure

Features of Fatigue Phenomenon

bull Second Stage Crack Propagation phasendashThis crack grows slowly under steady fatigue loading

but starts to accelerate as the net section decreases increasing the local stress at the crack-front

bull Third and Final Stage Failure ndashFailure occurs as an unstable fracture when the

remaining area is too small to support the load

hellip 2

bull Stages create distinctive features of the fracture surface - used to identify fatigue failure in failure investigations

5192016

7

Tell-tale marks of Fatigue - Striations

Features of Fatigue Phenomenon

bull Some materials exhibit endurance limits ie a stress below which the life is infinitendash Steels typical endurance limit - 40 -

60 of YS

ndash solutes (carbon nitrogen) present in the material pin the dislocations and prevent their motion at small strains

ndash Al alloys No endurance limits

ndash Related to the absence of dislocation-pinning solutes

hellip 3

bull At large Nf the lifetime is dominated by nucleation

ndash Therefore strengthening the surface (shot penning) is beneficial to delay crack nucleation and extend life

5192016

8

S

N

bull Typically load life and load life

bull Experimentally-determined ndash Rotating bending mc round specimens

ndash Electrohydraulic mc flat (or other) specimens

bull The practical limit of experimental testing has been 106 or 107 cycles due to impractical test times frequency limitations of machines and artefacts of higher frequencies

bull Adjustment for mean stress using Goodman Relation

Observations on S-N Curve

S

N

bull X-Axis No of Cycles On Log10 scale Log10 N

bull Y-Axis Stress ndash Either Stress Amplitude (Alternating stress) or Max

Stress

ndash Alternating Stress (ie Stress Amplitude) is the most dominant aspect of load that influences fatigue

ndash Either linear scale or Log10 scale

bull Curve is obtained for a given stress ratio R

bull R = (Min Stress Max Stress)

Observations on S-N Curve

5192016

9

S

N

bull S-N curve data is generally developed under Constant Amplitude ndash we will discuss later variable amplitude

bull The curve is dependent on many conditions ndash Loading ratio min load max load (R-ratio)

ndash Material

ndash stress concentrators

ndash surface finish

ndash frequency cyclic shape dwell etc hellip Not much effect unless in corrosive environment

Observations on S-N Curve

S

N

High cycle fatigue vs Low cycle fatigue

bull HCF Life usually more than 104 cycles ndash The distinction between high cycle fatigue and low

cycle fatigue is reflected in most design standards as high cycle fatigue design standards normally give S-N curves starting with N = 104 cycles

bull LCF Life usually less than 104 cycles ndash For low cycle fatigue design the stress range

concept is not immediately valid since typically the low cycle fatigue strength is governed by (large inelastic) strains

Arbitrary Could be 103

Arbitrary Could be 103

5192016

10

HCF and LCF

bull HCFndash Typical of rotating components machineriesndash 100 RPM = 6000 cycles per hr = 144x105 day or

taking 8 hrday operation5x107 yrndash Automobile crankshaft may accumulate 108 cycles in

100000 km runndash Reciprocating machineries springshellip etc etc

bull LCFndash Airplane Ground-Air-Ground Cyclesndash Shipsndash Vehicle Chassishellip

Various types of fatigue

bull Fretting fatiguendash Fatigue associated with or aided by repetitive rubbing

of surfaces causing wear and cracks also often accompanied by corrosion Such cracking provides sites for fatigue damage and decreases fatiguestrength of materials operating under cycling stress

bull Sonic fatigue (Acoustic fatigue) ndash Fatigue caused by acoustic vibrations acoustic

pressures engine noise can generate high stresses

bull Thermomechanical fatigue ndash Mechanical loads (cyclic) also accompanied by

thermal cycles

5192016

6

Features of Fatigue Phenomenon

bull A gradually progressive process

bull Slow development of damage in the early stage followed by rapid growth towards the end

bull First stage crack initiation phase

ndashMay occupy most of the life in most cases (eg nicely machined parts small cyclic loads )

ndashUsually confined to a small area of high local stress

ndashAdjacent areas with only slightly lower stresses may not see any fatigue damage

ndashusually several independent micro-cracks grow and coalesce to form one dominant crack

hellip 1

Dislocations Slip Microcrack Macrocrack Crack Propagation Failure

Features of Fatigue Phenomenon

bull Second Stage Crack Propagation phasendashThis crack grows slowly under steady fatigue loading

but starts to accelerate as the net section decreases increasing the local stress at the crack-front

bull Third and Final Stage Failure ndashFailure occurs as an unstable fracture when the

remaining area is too small to support the load

hellip 2

bull Stages create distinctive features of the fracture surface - used to identify fatigue failure in failure investigations

5192016

7

Tell-tale marks of Fatigue - Striations

Features of Fatigue Phenomenon

bull Some materials exhibit endurance limits ie a stress below which the life is infinitendash Steels typical endurance limit - 40 -

60 of YS

ndash solutes (carbon nitrogen) present in the material pin the dislocations and prevent their motion at small strains

ndash Al alloys No endurance limits

ndash Related to the absence of dislocation-pinning solutes

hellip 3

bull At large Nf the lifetime is dominated by nucleation

ndash Therefore strengthening the surface (shot penning) is beneficial to delay crack nucleation and extend life

5192016

8

S

N

bull Typically load life and load life

bull Experimentally-determined ndash Rotating bending mc round specimens

ndash Electrohydraulic mc flat (or other) specimens

bull The practical limit of experimental testing has been 106 or 107 cycles due to impractical test times frequency limitations of machines and artefacts of higher frequencies

bull Adjustment for mean stress using Goodman Relation

Observations on S-N Curve

S

N

bull X-Axis No of Cycles On Log10 scale Log10 N

bull Y-Axis Stress ndash Either Stress Amplitude (Alternating stress) or Max

Stress

ndash Alternating Stress (ie Stress Amplitude) is the most dominant aspect of load that influences fatigue

ndash Either linear scale or Log10 scale

bull Curve is obtained for a given stress ratio R

bull R = (Min Stress Max Stress)

Observations on S-N Curve

5192016

9

S

N

bull S-N curve data is generally developed under Constant Amplitude ndash we will discuss later variable amplitude

bull The curve is dependent on many conditions ndash Loading ratio min load max load (R-ratio)

ndash Material

ndash stress concentrators

ndash surface finish

ndash frequency cyclic shape dwell etc hellip Not much effect unless in corrosive environment

Observations on S-N Curve

S

N

High cycle fatigue vs Low cycle fatigue

bull HCF Life usually more than 104 cycles ndash The distinction between high cycle fatigue and low

cycle fatigue is reflected in most design standards as high cycle fatigue design standards normally give S-N curves starting with N = 104 cycles

bull LCF Life usually less than 104 cycles ndash For low cycle fatigue design the stress range

concept is not immediately valid since typically the low cycle fatigue strength is governed by (large inelastic) strains

Arbitrary Could be 103

Arbitrary Could be 103

5192016

10

HCF and LCF

bull HCFndash Typical of rotating components machineriesndash 100 RPM = 6000 cycles per hr = 144x105 day or

taking 8 hrday operation5x107 yrndash Automobile crankshaft may accumulate 108 cycles in

100000 km runndash Reciprocating machineries springshellip etc etc

bull LCFndash Airplane Ground-Air-Ground Cyclesndash Shipsndash Vehicle Chassishellip

Various types of fatigue

bull Fretting fatiguendash Fatigue associated with or aided by repetitive rubbing

of surfaces causing wear and cracks also often accompanied by corrosion Such cracking provides sites for fatigue damage and decreases fatiguestrength of materials operating under cycling stress

bull Sonic fatigue (Acoustic fatigue) ndash Fatigue caused by acoustic vibrations acoustic

pressures engine noise can generate high stresses

bull Thermomechanical fatigue ndash Mechanical loads (cyclic) also accompanied by

thermal cycles

5192016

7

Tell-tale marks of Fatigue - Striations

Features of Fatigue Phenomenon

bull Some materials exhibit endurance limits ie a stress below which the life is infinitendash Steels typical endurance limit - 40 -

60 of YS

ndash solutes (carbon nitrogen) present in the material pin the dislocations and prevent their motion at small strains

ndash Al alloys No endurance limits

ndash Related to the absence of dislocation-pinning solutes

hellip 3

bull At large Nf the lifetime is dominated by nucleation

ndash Therefore strengthening the surface (shot penning) is beneficial to delay crack nucleation and extend life

5192016

8

S

N

bull Typically load life and load life

bull Experimentally-determined ndash Rotating bending mc round specimens

ndash Electrohydraulic mc flat (or other) specimens

bull The practical limit of experimental testing has been 106 or 107 cycles due to impractical test times frequency limitations of machines and artefacts of higher frequencies

bull Adjustment for mean stress using Goodman Relation

Observations on S-N Curve

S

N

bull X-Axis No of Cycles On Log10 scale Log10 N

bull Y-Axis Stress ndash Either Stress Amplitude (Alternating stress) or Max

Stress

ndash Alternating Stress (ie Stress Amplitude) is the most dominant aspect of load that influences fatigue

ndash Either linear scale or Log10 scale

bull Curve is obtained for a given stress ratio R

bull R = (Min Stress Max Stress)

Observations on S-N Curve

5192016

9

S

N

bull S-N curve data is generally developed under Constant Amplitude ndash we will discuss later variable amplitude

bull The curve is dependent on many conditions ndash Loading ratio min load max load (R-ratio)

ndash Material

ndash stress concentrators

ndash surface finish

ndash frequency cyclic shape dwell etc hellip Not much effect unless in corrosive environment

Observations on S-N Curve

S

N

High cycle fatigue vs Low cycle fatigue

bull HCF Life usually more than 104 cycles ndash The distinction between high cycle fatigue and low

cycle fatigue is reflected in most design standards as high cycle fatigue design standards normally give S-N curves starting with N = 104 cycles

bull LCF Life usually less than 104 cycles ndash For low cycle fatigue design the stress range

concept is not immediately valid since typically the low cycle fatigue strength is governed by (large inelastic) strains

Arbitrary Could be 103

Arbitrary Could be 103

5192016

10

HCF and LCF

bull HCFndash Typical of rotating components machineriesndash 100 RPM = 6000 cycles per hr = 144x105 day or

taking 8 hrday operation5x107 yrndash Automobile crankshaft may accumulate 108 cycles in

100000 km runndash Reciprocating machineries springshellip etc etc

bull LCFndash Airplane Ground-Air-Ground Cyclesndash Shipsndash Vehicle Chassishellip

Various types of fatigue

bull Fretting fatiguendash Fatigue associated with or aided by repetitive rubbing

of surfaces causing wear and cracks also often accompanied by corrosion Such cracking provides sites for fatigue damage and decreases fatiguestrength of materials operating under cycling stress

bull Sonic fatigue (Acoustic fatigue) ndash Fatigue caused by acoustic vibrations acoustic

pressures engine noise can generate high stresses

bull Thermomechanical fatigue ndash Mechanical loads (cyclic) also accompanied by

thermal cycles

5192016

8

S

N

bull Typically load life and load life

bull Experimentally-determined ndash Rotating bending mc round specimens

ndash Electrohydraulic mc flat (or other) specimens

bull The practical limit of experimental testing has been 106 or 107 cycles due to impractical test times frequency limitations of machines and artefacts of higher frequencies

bull Adjustment for mean stress using Goodman Relation

Observations on S-N Curve

S

N

bull X-Axis No of Cycles On Log10 scale Log10 N

bull Y-Axis Stress ndash Either Stress Amplitude (Alternating stress) or Max

Stress

ndash Alternating Stress (ie Stress Amplitude) is the most dominant aspect of load that influences fatigue

ndash Either linear scale or Log10 scale

bull Curve is obtained for a given stress ratio R

bull R = (Min Stress Max Stress)

Observations on S-N Curve

5192016

9

S

N

bull S-N curve data is generally developed under Constant Amplitude ndash we will discuss later variable amplitude

bull The curve is dependent on many conditions ndash Loading ratio min load max load (R-ratio)

ndash Material

ndash stress concentrators

ndash surface finish

ndash frequency cyclic shape dwell etc hellip Not much effect unless in corrosive environment

Observations on S-N Curve

S

N

High cycle fatigue vs Low cycle fatigue

bull HCF Life usually more than 104 cycles ndash The distinction between high cycle fatigue and low

cycle fatigue is reflected in most design standards as high cycle fatigue design standards normally give S-N curves starting with N = 104 cycles

bull LCF Life usually less than 104 cycles ndash For low cycle fatigue design the stress range

concept is not immediately valid since typically the low cycle fatigue strength is governed by (large inelastic) strains

Arbitrary Could be 103

Arbitrary Could be 103

5192016

10

HCF and LCF

bull HCFndash Typical of rotating components machineriesndash 100 RPM = 6000 cycles per hr = 144x105 day or

taking 8 hrday operation5x107 yrndash Automobile crankshaft may accumulate 108 cycles in

100000 km runndash Reciprocating machineries springshellip etc etc

bull LCFndash Airplane Ground-Air-Ground Cyclesndash Shipsndash Vehicle Chassishellip

Various types of fatigue

bull Fretting fatiguendash Fatigue associated with or aided by repetitive rubbing

of surfaces causing wear and cracks also often accompanied by corrosion Such cracking provides sites for fatigue damage and decreases fatiguestrength of materials operating under cycling stress

bull Sonic fatigue (Acoustic fatigue) ndash Fatigue caused by acoustic vibrations acoustic

pressures engine noise can generate high stresses

bull Thermomechanical fatigue ndash Mechanical loads (cyclic) also accompanied by

thermal cycles

5192016

9

S

N

bull S-N curve data is generally developed under Constant Amplitude ndash we will discuss later variable amplitude

bull The curve is dependent on many conditions ndash Loading ratio min load max load (R-ratio)

ndash Material

ndash stress concentrators

ndash surface finish

ndash frequency cyclic shape dwell etc hellip Not much effect unless in corrosive environment

Observations on S-N Curve

S

N

High cycle fatigue vs Low cycle fatigue

bull HCF Life usually more than 104 cycles ndash The distinction between high cycle fatigue and low

cycle fatigue is reflected in most design standards as high cycle fatigue design standards normally give S-N curves starting with N = 104 cycles

bull LCF Life usually less than 104 cycles ndash For low cycle fatigue design the stress range

concept is not immediately valid since typically the low cycle fatigue strength is governed by (large inelastic) strains

Arbitrary Could be 103

Arbitrary Could be 103

5192016

10

HCF and LCF

bull HCFndash Typical of rotating components machineriesndash 100 RPM = 6000 cycles per hr = 144x105 day or

taking 8 hrday operation5x107 yrndash Automobile crankshaft may accumulate 108 cycles in

100000 km runndash Reciprocating machineries springshellip etc etc

bull LCFndash Airplane Ground-Air-Ground Cyclesndash Shipsndash Vehicle Chassishellip

Various types of fatigue

bull Fretting fatiguendash Fatigue associated with or aided by repetitive rubbing

of surfaces causing wear and cracks also often accompanied by corrosion Such cracking provides sites for fatigue damage and decreases fatiguestrength of materials operating under cycling stress

bull Sonic fatigue (Acoustic fatigue) ndash Fatigue caused by acoustic vibrations acoustic

pressures engine noise can generate high stresses

bull Thermomechanical fatigue ndash Mechanical loads (cyclic) also accompanied by

thermal cycles

5192016

10

HCF and LCF

bull HCFndash Typical of rotating components machineriesndash 100 RPM = 6000 cycles per hr = 144x105 day or

taking 8 hrday operation5x107 yrndash Automobile crankshaft may accumulate 108 cycles in

100000 km runndash Reciprocating machineries springshellip etc etc

bull LCFndash Airplane Ground-Air-Ground Cyclesndash Shipsndash Vehicle Chassishellip

Various types of fatigue

bull Fretting fatiguendash Fatigue associated with or aided by repetitive rubbing

of surfaces causing wear and cracks also often accompanied by corrosion Such cracking provides sites for fatigue damage and decreases fatiguestrength of materials operating under cycling stress

bull Sonic fatigue (Acoustic fatigue) ndash Fatigue caused by acoustic vibrations acoustic

pressures engine noise can generate high stresses

bull Thermomechanical fatigue ndash Mechanical loads (cyclic) also accompanied by

thermal cycles

5192016

11

Endurance Limit Fatigue Limit Fatigue Strength

All these terms refer to ldquohow much stress amplitude can be applied in a cyclic loading without causing a failurerdquobull Endurance Limit A stress level below which a material can

be cycled indefinitely without failurendash Several Steels Ti-alloys Mo-alloys etc exhibit such endurance

limit

Many materials (eg Al-alloys Mg Cu Ni Alloys even some steels) do not have Endurance Limit For such materials we define bull Fatigue Strength as the max stress amplitude that can be

applied to get given life (say 106 cycles 5x108 cycles etc)bull Fatigue Limit as the limiting value of stress amplitude at

which the life (no of cycles to failure) becomes ldquovery largerdquo

[Theoretically there is no ldquoproofrdquo of endurance limithellipNo infinite lifehellip it may be just that we havenrsquot tested for large number of cycles So Fatigue Limit is a more realistic term]

22

Axial loading

S-N Curve Fatigue Tests Test Arrangements Rotating Bending loading

5192016

12

Fatigue Strengths for SteelsRotating bending 107 or 108 Cycles

Tensile Strength Su ksi

Alt

Fat

igu

e St

ren

gth

Sf

ksi

MP

a

MPa

Ref Ali Fatemi eFatigue Chap4

x Carbon Steels Alloy Steels

Fatigue Strengths for Al-alloysRotating bending 108 Cycles

Tensile Strength Su ksi

Alt

Fat

igu

e St

ren

gth

Sf

ksi

MP

a

MPa

Ref Ali Fatemi eFatigue Chap4

x Wrought Cast

Metal Fatigue in Engineering by Ralph I Stephens Ali Fatemi Robert R Stephens Henry O Fuchs John Wiley amp Sons Nov 2000

5192016

12

Fatigue Strengths for SteelsRotating bending 107 or 108 Cycles

Tensile Strength Su ksi

Alt

Fat

igu

e St

ren

gth

Sf

ksi

MP

a

MPa

Ref Ali Fatemi eFatigue Chap4

x Carbon Steels Alloy Steels

Fatigue Strengths for Al-alloysRotating bending 108 Cycles

Tensile Strength Su ksi

Alt

Fat

igu

e St

ren

gth

Sf

ksi

MP

a

MPa

Ref Ali Fatemi eFatigue Chap4

x Wrought Cast

Metal Fatigue in Engineering by Ralph I Stephens Ali Fatemi Robert R Stephens Henry O Fuchs John Wiley amp Sons Nov 2000

5192016

13

Some related ASTM test standards

Designation Title

E466 - 15Standard Practice for Conducting Force Controlled Constant Amplitude Axial Fatigue Tests of Metallic Materials

E468 - 11Standard Practice for Presentation of Constant Amplitude Fatigue Test Results for Metallic Materials

E606 E606M - 12

Standard Test Method for Strain-Controlled Fatigue Testing

E2714 - 13 Standard Test Method for Creep-Fatigue Testing

E2789 -10(2015)

Standard Guide for Fretting Fatigue Testing

E2948 - 14Standard Test Method for Conducting Rotating Bending Fatigue Tests of Solid Round Fine Wire

Cyclic Deformation and Fatigue Crack Formation

E739 -10(2015)

Standard Practice for Statistical Analysis of Linear or Linearized Stress-Life (S-N) and Strain-Life (ε-N) Fatigue Data

E1049 -85(2011)e1

Standard Practices for Cycle Counting in Fatigue Analysis

log 119878119886 = log 119886 + 119887 log119873there4 119878119886 = 119886119873119887

bull Often N is written as 2Nf

which is no of reversals

bull Constant a is generally representative of the ldquoFracture Strengthrdquo of material say 119878119891

prime (also called ldquoFatigue Strength Coeffrdquo)

there4 119878119886 = 119878119891prime 2119873119891

119887

119874119877 119878119886 = 119878119891prime119873119887

bull b is Basquin (Year 1910) Exponent

bull b -005 to -015

Observations on S-N Curve

Sa

Log N

Log N

log Sa

1

b

BasquinRelation

5192016

14

Nomenclature in Cyclic Loading

120590119898119886119909

120590119898119894119899

120590119898

120590119886 ∆120590

0

∆120590 ∶ 119878119905119903119890119904119904 119877119886119899119892119890 = 120590119898119886119909 minus 120590119898119894119899

120590119886 119860119897119905119890119903119899119886119905119894119899119892 119878119905119903119890119904119904 = ∆1205902 = (120590119898119886119909minus120590119898119894119899)2

120590119898 119872119890119886119899 119878119905119903119890119904119904 = (120590119898119886119909+120590119898119894119899)2

119877 119878119905119903119890119904119904 119877119886119905119894119900 =120590119898119894119899

120590119898119886119909

120590119898119894119899 119872119894119899119894119898119906119898 119878119905119903119890119904119904

120590119898119886119909119872119886119909119894119898119906119898 119878119905119903119890119904119904

Constant Amplitude Fatigue Loading

120590119898119886119909 = 100 120590119898119894119899= minus100120590119898 = 0 120590119886 = 100119877119886119899119892119890 = 200 119877 = minus1

120590119898119886119909 = 100 120590119898119894119899= 20120590119898 = 60 120590119886 = 40119877119886119899119892119890 = 80 119877 = 02

120590119898119886119909 = minus10 120590119898119894119899= minus120120590119898 = minus65 120590119886 = 55119877119886119899119892119890 = 110 119877 = 12

0

100

minus100

Tension - Compression

0

100

20

Tension - Tension

0minus10

minus120

Compression - Compression

5192016

15

Constant Amplitude Fatigue Loading

Comp ndash Comp R gt 1

0 ndash Comp R = infin

Tension ndash Comp R = -1

0 ndash TensionR = 0

Tension - Tension 0 lt R lt 1

Time

Stress

Effect of Mean Stress

5192016

16

Mean Stress Effect

Sm = 0

Stre

ss A

mp

litu

de

Sa

Life Cycles N

bull The damaging portion of load cycle Tensile stresso this causes cracks to open and grow

bull Baseline Sa-N curve Zero mean stress (Sm=0) and cyclic stress amplitude (alternating stress) Sa

bull Let us superimpose an additional steady stress Sm on this

Sm lt 0

bull If Smlt0 (ie Compr) Max Tens Stress and N The curve shifts to right

Sm gt 0

bull If Smgt0 (ie tensile) Max Tens Stress and N The curve shifts to left

Goodman Relation

Sm = 0

Stre

ss A

mp

litu

de

Sa

Life Cycles N

Sa0

N

1

1

3

3

4

4Constant Life N

SmSu

SaSa0

1

1

2

2

Sm gt 0

Su

m = 1

m lt 1

m gt 1

119878119886

1198781198860+

119878119898

119878119906

119898

= 1

In general m ~ 06 ndash 2For most calculations we assume m=1

5192016

17

Mean Stress Effect Goodman amp Others

httpwwwfgguni-ljsi~pmozeesdepmasterwg12l0200htm

Goodman Gerber Soderberg LinesFailure envelopes

Gerber Parabola(m = 2)

5192016

18

Goodman Diagram Constant Life Lines

Constructing failure envelopes using Goodman Diagram

Constant Life Lines

Constant Load Lines

httpneilwimerweeblycombasic-fatigue-analysishtml

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

5192016

19

Effect of R-ratio

Al Alloy 2024 T3Axial Loading

S max

ksi

N

102 103 104 105 106 107 108

40

60

0

80

20

73 ksi = UTS

R= SminSmax

+060

+050

+040

+025

+010

0

-030

-10

Ref Fatiue of Aircraft Structures NAVAIR 01-1A-13 1966

Questions

5192016

20

Numerical Example 1

In a Tension-Compression (R=-1) fatigue test with an alloy following mean fatigue lives were found for various stress levels

The ultimate strength of the alloy was found to be 462 MPa(i) Plot the Sa-N curve (ii) Check whether this obeys Basquin relation Find

Basquin exponent(iii) Using Goodman relation derive the Sa- N curve for

R=0

Smax (MPa) 138 172 207 276 345 380

N 12589250 1273500 199526 15850 1778 1060

Solution

0

100

200

300

400

10E+02 10E+04 10E+06 10E+08

Sa vs N

0

05

1

15

2

25

3

0 2 4 6 8

log Sa vs log N

Since R=-1 Sa=Smax

Basquin Exponent -0107Sfrsquo = 102892 = 780 MPa

WE take m=1 For R=0 Sm=Sa so that the eqn is 119878119886

1198781198860+

119878119898

119878119906

119898

= 1

119878119886

1198781198860+

119878119886

119878119906= 1rArr 119904119886 =

11

1198781198860+

1

119878119906

Substituting values of Sa0 we get Sa for each value of N Then we can plot

5192016

21

Solutionhellip continued

N 12589250 1273500 199526 15850 1778 1060

Smax (MPa) 138 172 207 276 345 380

Sa0 (R=-1) 138 172 207 276 345 380

Sa (R=0) 106 125 143 173 198 209

0

50

100

150

200

250

300

350

400

10E+02 10E+03 10E+04 10E+05 10E+06 10E+07 10E+08

Sa vs N

R = -1

R = 0

Numerical Example 2

For an Aluminium Alloy the S-N behaviour in tension compression fatigue with zero mean stress is approximated by the relation

119878119886= 480119873minus012

(i) What would be the life of a structure made of this alloy if it is subjected to cyclic stress of 80 MPa

(ii) What would be life if there is an additional steady stress of +10 MPa

(iii) What would be life if the additional steady stress in -10 MPa

5192016

22

Solution to Numerical Example 2

(i) This is direct substitution in the given stress-life curve equationWe have Sa = 80 MPa Therefore

80 = 480119873minus012

This gives N = (48080)^(1012) = 3052065

(ii) With an additional steady stress of +10 MPaHere the mean stress is now Sm=10 MPa and Sa=80 MPaSince mean stress is not zero we need to use Goodman relation to arrive at the mean stress effect

With mean stress zero 1198781198860= 119878119891prime119873119887

Recall Goodman relation

In absence data we take Su = Sfrsquo Now we can just substitute for Sa(=80) and Sm (=10) and Su=Sfrsquo = 480 with b= -012 and get N

Ans N= 2560937

(iii) Similarly with Sm = -10 we can repeat calculation

Ans 3624244

119878119886

1198781198860+

119878119898

119878119906= 1rArr

119878119886

119878119891prime119873119887 +

119878119898

119878119906= 1

Stress ConcentrationHole in a large plate under uniaxial tension

bull Large plate Max stress is 3 times the nominal stress (applied stress far-field stress)

bull Stress concentration Factor Kt = SmaxSn = 3

Elliptic hole SCF = 1+ 2(ba) = 1+2(a)

smax

sns

5192016

23

Stress Concentration (contd)

Petersonrsquos Original

compilationndash updated and

enhanced

5192016

24

Fatigue under Stress Concentration

S

N

s

s Krsquo (KrsquoltKt)

s Kt

Notched

Un-notched

Effective Stress Concentration for fatigue is less than the Stress Concentration in Static

Many types of discontinuities which act as stress concentrators Holes cut-outs key-slots cracks notches hellip

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119904119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ

119870119891 = 1 + 119902 119870119905 minus 1

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

Fatigue notch sensitivity factor

Stress Concentration Factor

119870119905 =119872119886119909 119904119905119903119890119904119904 119908119894119905ℎ 119899119900119905119888ℎ

119878119905119903119890119904119904 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ 1 lt 119870119891 lt 119870119905

Fatigue Notch Sensitivity Factor q119870119891 = 1 + 119902 119870119905 minus 1 and

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

5192016

25

Estimates of 119902 ∶ All Empirical

bull Neuberrsquos Approx Formula (R=-1) 119954 =120783

120783+ Τ120646 119955(r = notch root radius 120588 = material characteristic length)bull For aluminium alloys

Ult Strength 150 300 600 MPa120588 20 06 04 mm

bull Petersonrsquos Approx Formula (R=-1) 119954 =120783

120783+119938

119955(r = notch root radius 119886 = another mat characteristic length)

bull For Steels 119886 = 002542070

119878119906

18

(Su in MPa a in mm)

bull For Al-alloys 119886 = 0635 119898119898 119886119901119901119903119900119909

NeuberParameter

0

005

01

015

02

025

03

0 500 1000 1500

Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500 1000 1500

Ne

ub

er

Par

ame

ter

Su MPa

Al-Alloys

Steels

5192016

26

Goodman Diagram for notched fatigue

bull Rather simplisticbull Does not

account for variation of Kf

with mean stress or life cycles

S a

Sa

Accounting for means stress effectHaig Diagrams

5192016

27

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

More about notched fatiguehellip

bull Mean stress effect is more pronounced in notched specimens than in smooth ones

bull Tensile mean stress accentuates the notch effect in fatigue hellip in fact Kf can exceed Kt in some cases Need to be catered for otherwise fatal in fatigue loading

bull Compressive mean stress can be beneficial It significantly reduces the effect of stress concentration and may in fact eliminate it

bull Residual stresses can contribute significantly to the level of mean stress This in turn may affect the notched behaviour

5192016

28

Questions

5192016

15

Constant Amplitude Fatigue Loading

Comp ndash Comp R gt 1

0 ndash Comp R = infin

Tension ndash Comp R = -1

0 ndash TensionR = 0

Tension - Tension 0 lt R lt 1

Time

Stress

Effect of Mean Stress

5192016

16

Mean Stress Effect

Sm = 0

Stre

ss A

mp

litu

de

Sa

Life Cycles N

bull The damaging portion of load cycle Tensile stresso this causes cracks to open and grow

bull Baseline Sa-N curve Zero mean stress (Sm=0) and cyclic stress amplitude (alternating stress) Sa

bull Let us superimpose an additional steady stress Sm on this

Sm lt 0

bull If Smlt0 (ie Compr) Max Tens Stress and N The curve shifts to right

Sm gt 0

bull If Smgt0 (ie tensile) Max Tens Stress and N The curve shifts to left

Goodman Relation

Sm = 0

Stre

ss A

mp

litu

de

Sa

Life Cycles N

Sa0

N

1

1

3

3

4

4Constant Life N

SmSu

SaSa0

1

1

2

2

Sm gt 0

Su

m = 1

m lt 1

m gt 1

119878119886

1198781198860+

119878119898

119878119906

119898

= 1

In general m ~ 06 ndash 2For most calculations we assume m=1

5192016

17

Mean Stress Effect Goodman amp Others

httpwwwfgguni-ljsi~pmozeesdepmasterwg12l0200htm

Goodman Gerber Soderberg LinesFailure envelopes

Gerber Parabola(m = 2)

5192016

18

Goodman Diagram Constant Life Lines

Constructing failure envelopes using Goodman Diagram

Constant Life Lines

Constant Load Lines

httpneilwimerweeblycombasic-fatigue-analysishtml

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

5192016

19

Effect of R-ratio

Al Alloy 2024 T3Axial Loading

S max

ksi

N

102 103 104 105 106 107 108

40

60

0

80

20

73 ksi = UTS

R= SminSmax

+060

+050

+040

+025

+010

0

-030

-10

Ref Fatiue of Aircraft Structures NAVAIR 01-1A-13 1966

Questions

5192016

20

Numerical Example 1

In a Tension-Compression (R=-1) fatigue test with an alloy following mean fatigue lives were found for various stress levels

The ultimate strength of the alloy was found to be 462 MPa(i) Plot the Sa-N curve (ii) Check whether this obeys Basquin relation Find

Basquin exponent(iii) Using Goodman relation derive the Sa- N curve for

R=0

Smax (MPa) 138 172 207 276 345 380

N 12589250 1273500 199526 15850 1778 1060

Solution

0

100

200

300

400

10E+02 10E+04 10E+06 10E+08

Sa vs N

0

05

1

15

2

25

3

0 2 4 6 8

log Sa vs log N

Since R=-1 Sa=Smax

Basquin Exponent -0107Sfrsquo = 102892 = 780 MPa

WE take m=1 For R=0 Sm=Sa so that the eqn is 119878119886

1198781198860+

119878119898

119878119906

119898

= 1

119878119886

1198781198860+

119878119886

119878119906= 1rArr 119904119886 =

11

1198781198860+

1

119878119906

Substituting values of Sa0 we get Sa for each value of N Then we can plot

5192016

21

Solutionhellip continued

N 12589250 1273500 199526 15850 1778 1060

Smax (MPa) 138 172 207 276 345 380

Sa0 (R=-1) 138 172 207 276 345 380

Sa (R=0) 106 125 143 173 198 209

0

50

100

150

200

250

300

350

400

10E+02 10E+03 10E+04 10E+05 10E+06 10E+07 10E+08

Sa vs N

R = -1

R = 0

Numerical Example 2

For an Aluminium Alloy the S-N behaviour in tension compression fatigue with zero mean stress is approximated by the relation

119878119886= 480119873minus012

(i) What would be the life of a structure made of this alloy if it is subjected to cyclic stress of 80 MPa

(ii) What would be life if there is an additional steady stress of +10 MPa

(iii) What would be life if the additional steady stress in -10 MPa

5192016

22

Solution to Numerical Example 2

(i) This is direct substitution in the given stress-life curve equationWe have Sa = 80 MPa Therefore

80 = 480119873minus012

This gives N = (48080)^(1012) = 3052065

(ii) With an additional steady stress of +10 MPaHere the mean stress is now Sm=10 MPa and Sa=80 MPaSince mean stress is not zero we need to use Goodman relation to arrive at the mean stress effect

With mean stress zero 1198781198860= 119878119891prime119873119887

Recall Goodman relation

In absence data we take Su = Sfrsquo Now we can just substitute for Sa(=80) and Sm (=10) and Su=Sfrsquo = 480 with b= -012 and get N

Ans N= 2560937

(iii) Similarly with Sm = -10 we can repeat calculation

Ans 3624244

119878119886

1198781198860+

119878119898

119878119906= 1rArr

119878119886

119878119891prime119873119887 +

119878119898

119878119906= 1

Stress ConcentrationHole in a large plate under uniaxial tension

bull Large plate Max stress is 3 times the nominal stress (applied stress far-field stress)

bull Stress concentration Factor Kt = SmaxSn = 3

Elliptic hole SCF = 1+ 2(ba) = 1+2(a)

smax

sns

5192016

23

Stress Concentration (contd)

Petersonrsquos Original

compilationndash updated and

enhanced

5192016

24

Fatigue under Stress Concentration

S

N

s

s Krsquo (KrsquoltKt)

s Kt

Notched

Un-notched

Effective Stress Concentration for fatigue is less than the Stress Concentration in Static

Many types of discontinuities which act as stress concentrators Holes cut-outs key-slots cracks notches hellip

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119904119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ

119870119891 = 1 + 119902 119870119905 minus 1

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

Fatigue notch sensitivity factor

Stress Concentration Factor

119870119905 =119872119886119909 119904119905119903119890119904119904 119908119894119905ℎ 119899119900119905119888ℎ

119878119905119903119890119904119904 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ 1 lt 119870119891 lt 119870119905

Fatigue Notch Sensitivity Factor q119870119891 = 1 + 119902 119870119905 minus 1 and

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

5192016

25

Estimates of 119902 ∶ All Empirical

bull Neuberrsquos Approx Formula (R=-1) 119954 =120783

120783+ Τ120646 119955(r = notch root radius 120588 = material characteristic length)bull For aluminium alloys

Ult Strength 150 300 600 MPa120588 20 06 04 mm

bull Petersonrsquos Approx Formula (R=-1) 119954 =120783

120783+119938

119955(r = notch root radius 119886 = another mat characteristic length)

bull For Steels 119886 = 002542070

119878119906

18

(Su in MPa a in mm)

bull For Al-alloys 119886 = 0635 119898119898 119886119901119901119903119900119909

NeuberParameter

0

005

01

015

02

025

03

0 500 1000 1500

Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500 1000 1500

Ne

ub

er

Par

ame

ter

Su MPa

Al-Alloys

Steels

5192016

26

Goodman Diagram for notched fatigue

bull Rather simplisticbull Does not

account for variation of Kf

with mean stress or life cycles

S a

Sa

Accounting for means stress effectHaig Diagrams

5192016

27

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

More about notched fatiguehellip

bull Mean stress effect is more pronounced in notched specimens than in smooth ones

bull Tensile mean stress accentuates the notch effect in fatigue hellip in fact Kf can exceed Kt in some cases Need to be catered for otherwise fatal in fatigue loading

bull Compressive mean stress can be beneficial It significantly reduces the effect of stress concentration and may in fact eliminate it

bull Residual stresses can contribute significantly to the level of mean stress This in turn may affect the notched behaviour

5192016

28

Questions

5192016

16

Mean Stress Effect

Sm = 0

Stre

ss A

mp

litu

de

Sa

Life Cycles N

bull The damaging portion of load cycle Tensile stresso this causes cracks to open and grow

bull Baseline Sa-N curve Zero mean stress (Sm=0) and cyclic stress amplitude (alternating stress) Sa

bull Let us superimpose an additional steady stress Sm on this

Sm lt 0

bull If Smlt0 (ie Compr) Max Tens Stress and N The curve shifts to right

Sm gt 0

bull If Smgt0 (ie tensile) Max Tens Stress and N The curve shifts to left

Goodman Relation

Sm = 0

Stre

ss A

mp

litu

de

Sa

Life Cycles N

Sa0

N

1

1

3

3

4

4Constant Life N

SmSu

SaSa0

1

1

2

2

Sm gt 0

Su

m = 1

m lt 1

m gt 1

119878119886

1198781198860+

119878119898

119878119906

119898

= 1

In general m ~ 06 ndash 2For most calculations we assume m=1

5192016

17

Mean Stress Effect Goodman amp Others

httpwwwfgguni-ljsi~pmozeesdepmasterwg12l0200htm

Goodman Gerber Soderberg LinesFailure envelopes

Gerber Parabola(m = 2)

5192016

18

Goodman Diagram Constant Life Lines

Constructing failure envelopes using Goodman Diagram

Constant Life Lines

Constant Load Lines

httpneilwimerweeblycombasic-fatigue-analysishtml

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

5192016

19

Effect of R-ratio

Al Alloy 2024 T3Axial Loading

S max

ksi

N

102 103 104 105 106 107 108

40

60

0

80

20

73 ksi = UTS

R= SminSmax

+060

+050

+040

+025

+010

0

-030

-10

Ref Fatiue of Aircraft Structures NAVAIR 01-1A-13 1966

Questions

5192016

20

Numerical Example 1

In a Tension-Compression (R=-1) fatigue test with an alloy following mean fatigue lives were found for various stress levels

The ultimate strength of the alloy was found to be 462 MPa(i) Plot the Sa-N curve (ii) Check whether this obeys Basquin relation Find

Basquin exponent(iii) Using Goodman relation derive the Sa- N curve for

R=0

Smax (MPa) 138 172 207 276 345 380

N 12589250 1273500 199526 15850 1778 1060

Solution

0

100

200

300

400

10E+02 10E+04 10E+06 10E+08

Sa vs N

0

05

1

15

2

25

3

0 2 4 6 8

log Sa vs log N

Since R=-1 Sa=Smax

Basquin Exponent -0107Sfrsquo = 102892 = 780 MPa

WE take m=1 For R=0 Sm=Sa so that the eqn is 119878119886

1198781198860+

119878119898

119878119906

119898

= 1

119878119886

1198781198860+

119878119886

119878119906= 1rArr 119904119886 =

11

1198781198860+

1

119878119906

Substituting values of Sa0 we get Sa for each value of N Then we can plot

5192016

21

Solutionhellip continued

N 12589250 1273500 199526 15850 1778 1060

Smax (MPa) 138 172 207 276 345 380

Sa0 (R=-1) 138 172 207 276 345 380

Sa (R=0) 106 125 143 173 198 209

0

50

100

150

200

250

300

350

400

10E+02 10E+03 10E+04 10E+05 10E+06 10E+07 10E+08

Sa vs N

R = -1

R = 0

Numerical Example 2

For an Aluminium Alloy the S-N behaviour in tension compression fatigue with zero mean stress is approximated by the relation

119878119886= 480119873minus012

(i) What would be the life of a structure made of this alloy if it is subjected to cyclic stress of 80 MPa

(ii) What would be life if there is an additional steady stress of +10 MPa

(iii) What would be life if the additional steady stress in -10 MPa

5192016

22

Solution to Numerical Example 2

(i) This is direct substitution in the given stress-life curve equationWe have Sa = 80 MPa Therefore

80 = 480119873minus012

This gives N = (48080)^(1012) = 3052065

(ii) With an additional steady stress of +10 MPaHere the mean stress is now Sm=10 MPa and Sa=80 MPaSince mean stress is not zero we need to use Goodman relation to arrive at the mean stress effect

With mean stress zero 1198781198860= 119878119891prime119873119887

Recall Goodman relation

In absence data we take Su = Sfrsquo Now we can just substitute for Sa(=80) and Sm (=10) and Su=Sfrsquo = 480 with b= -012 and get N

Ans N= 2560937

(iii) Similarly with Sm = -10 we can repeat calculation

Ans 3624244

119878119886

1198781198860+

119878119898

119878119906= 1rArr

119878119886

119878119891prime119873119887 +

119878119898

119878119906= 1

Stress ConcentrationHole in a large plate under uniaxial tension

bull Large plate Max stress is 3 times the nominal stress (applied stress far-field stress)

bull Stress concentration Factor Kt = SmaxSn = 3

Elliptic hole SCF = 1+ 2(ba) = 1+2(a)

smax

sns

5192016

23

Stress Concentration (contd)

Petersonrsquos Original

compilationndash updated and

enhanced

5192016

24

Fatigue under Stress Concentration

S

N

s

s Krsquo (KrsquoltKt)

s Kt

Notched

Un-notched

Effective Stress Concentration for fatigue is less than the Stress Concentration in Static

Many types of discontinuities which act as stress concentrators Holes cut-outs key-slots cracks notches hellip

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119904119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ

119870119891 = 1 + 119902 119870119905 minus 1

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

Fatigue notch sensitivity factor

Stress Concentration Factor

119870119905 =119872119886119909 119904119905119903119890119904119904 119908119894119905ℎ 119899119900119905119888ℎ

119878119905119903119890119904119904 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ 1 lt 119870119891 lt 119870119905

Fatigue Notch Sensitivity Factor q119870119891 = 1 + 119902 119870119905 minus 1 and

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

5192016

25

Estimates of 119902 ∶ All Empirical

bull Neuberrsquos Approx Formula (R=-1) 119954 =120783

120783+ Τ120646 119955(r = notch root radius 120588 = material characteristic length)bull For aluminium alloys

Ult Strength 150 300 600 MPa120588 20 06 04 mm

bull Petersonrsquos Approx Formula (R=-1) 119954 =120783

120783+119938

119955(r = notch root radius 119886 = another mat characteristic length)

bull For Steels 119886 = 002542070

119878119906

18

(Su in MPa a in mm)

bull For Al-alloys 119886 = 0635 119898119898 119886119901119901119903119900119909

NeuberParameter

0

005

01

015

02

025

03

0 500 1000 1500

Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500 1000 1500

Ne

ub

er

Par

ame

ter

Su MPa

Al-Alloys

Steels

5192016

26

Goodman Diagram for notched fatigue

bull Rather simplisticbull Does not

account for variation of Kf

with mean stress or life cycles

S a

Sa

Accounting for means stress effectHaig Diagrams

5192016

27

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

More about notched fatiguehellip

bull Mean stress effect is more pronounced in notched specimens than in smooth ones

bull Tensile mean stress accentuates the notch effect in fatigue hellip in fact Kf can exceed Kt in some cases Need to be catered for otherwise fatal in fatigue loading

bull Compressive mean stress can be beneficial It significantly reduces the effect of stress concentration and may in fact eliminate it

bull Residual stresses can contribute significantly to the level of mean stress This in turn may affect the notched behaviour

5192016

28

Questions

5192016

17

Mean Stress Effect Goodman amp Others

httpwwwfgguni-ljsi~pmozeesdepmasterwg12l0200htm

Goodman Gerber Soderberg LinesFailure envelopes

Gerber Parabola(m = 2)

5192016

18

Goodman Diagram Constant Life Lines

Constructing failure envelopes using Goodman Diagram

Constant Life Lines

Constant Load Lines

httpneilwimerweeblycombasic-fatigue-analysishtml

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

5192016

19

Effect of R-ratio

Al Alloy 2024 T3Axial Loading

S max

ksi

N

102 103 104 105 106 107 108

40

60

0

80

20

73 ksi = UTS

R= SminSmax

+060

+050

+040

+025

+010

0

-030

-10

Ref Fatiue of Aircraft Structures NAVAIR 01-1A-13 1966

Questions

5192016

20

Numerical Example 1

In a Tension-Compression (R=-1) fatigue test with an alloy following mean fatigue lives were found for various stress levels

The ultimate strength of the alloy was found to be 462 MPa(i) Plot the Sa-N curve (ii) Check whether this obeys Basquin relation Find

Basquin exponent(iii) Using Goodman relation derive the Sa- N curve for

R=0

Smax (MPa) 138 172 207 276 345 380

N 12589250 1273500 199526 15850 1778 1060

Solution

0

100

200

300

400

10E+02 10E+04 10E+06 10E+08

Sa vs N

0

05

1

15

2

25

3

0 2 4 6 8

log Sa vs log N

Since R=-1 Sa=Smax

Basquin Exponent -0107Sfrsquo = 102892 = 780 MPa

WE take m=1 For R=0 Sm=Sa so that the eqn is 119878119886

1198781198860+

119878119898

119878119906

119898

= 1

119878119886

1198781198860+

119878119886

119878119906= 1rArr 119904119886 =

11

1198781198860+

1

119878119906

Substituting values of Sa0 we get Sa for each value of N Then we can plot

5192016

21

Solutionhellip continued

N 12589250 1273500 199526 15850 1778 1060

Smax (MPa) 138 172 207 276 345 380

Sa0 (R=-1) 138 172 207 276 345 380

Sa (R=0) 106 125 143 173 198 209

0

50

100

150

200

250

300

350

400

10E+02 10E+03 10E+04 10E+05 10E+06 10E+07 10E+08

Sa vs N

R = -1

R = 0

Numerical Example 2

For an Aluminium Alloy the S-N behaviour in tension compression fatigue with zero mean stress is approximated by the relation

119878119886= 480119873minus012

(i) What would be the life of a structure made of this alloy if it is subjected to cyclic stress of 80 MPa

(ii) What would be life if there is an additional steady stress of +10 MPa

(iii) What would be life if the additional steady stress in -10 MPa

5192016

22

Solution to Numerical Example 2

(i) This is direct substitution in the given stress-life curve equationWe have Sa = 80 MPa Therefore

80 = 480119873minus012

This gives N = (48080)^(1012) = 3052065

(ii) With an additional steady stress of +10 MPaHere the mean stress is now Sm=10 MPa and Sa=80 MPaSince mean stress is not zero we need to use Goodman relation to arrive at the mean stress effect

With mean stress zero 1198781198860= 119878119891prime119873119887

Recall Goodman relation

In absence data we take Su = Sfrsquo Now we can just substitute for Sa(=80) and Sm (=10) and Su=Sfrsquo = 480 with b= -012 and get N

Ans N= 2560937

(iii) Similarly with Sm = -10 we can repeat calculation

Ans 3624244

119878119886

1198781198860+

119878119898

119878119906= 1rArr

119878119886

119878119891prime119873119887 +

119878119898

119878119906= 1

Stress ConcentrationHole in a large plate under uniaxial tension

bull Large plate Max stress is 3 times the nominal stress (applied stress far-field stress)

bull Stress concentration Factor Kt = SmaxSn = 3

Elliptic hole SCF = 1+ 2(ba) = 1+2(a)

smax

sns

5192016

23

Stress Concentration (contd)

Petersonrsquos Original

compilationndash updated and

enhanced

5192016

24

Fatigue under Stress Concentration

S

N

s

s Krsquo (KrsquoltKt)

s Kt

Notched

Un-notched

Effective Stress Concentration for fatigue is less than the Stress Concentration in Static

Many types of discontinuities which act as stress concentrators Holes cut-outs key-slots cracks notches hellip

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119904119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ

119870119891 = 1 + 119902 119870119905 minus 1

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

Fatigue notch sensitivity factor

Stress Concentration Factor

119870119905 =119872119886119909 119904119905119903119890119904119904 119908119894119905ℎ 119899119900119905119888ℎ

119878119905119903119890119904119904 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ 1 lt 119870119891 lt 119870119905

Fatigue Notch Sensitivity Factor q119870119891 = 1 + 119902 119870119905 minus 1 and

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

5192016

25

Estimates of 119902 ∶ All Empirical

bull Neuberrsquos Approx Formula (R=-1) 119954 =120783

120783+ Τ120646 119955(r = notch root radius 120588 = material characteristic length)bull For aluminium alloys

Ult Strength 150 300 600 MPa120588 20 06 04 mm

bull Petersonrsquos Approx Formula (R=-1) 119954 =120783

120783+119938

119955(r = notch root radius 119886 = another mat characteristic length)

bull For Steels 119886 = 002542070

119878119906

18

(Su in MPa a in mm)

bull For Al-alloys 119886 = 0635 119898119898 119886119901119901119903119900119909

NeuberParameter

0

005

01

015

02

025

03

0 500 1000 1500

Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500 1000 1500

Ne

ub

er

Par

ame

ter

Su MPa

Al-Alloys

Steels

5192016

26

Goodman Diagram for notched fatigue

bull Rather simplisticbull Does not

account for variation of Kf

with mean stress or life cycles

S a

Sa

Accounting for means stress effectHaig Diagrams

5192016

27

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

More about notched fatiguehellip

bull Mean stress effect is more pronounced in notched specimens than in smooth ones

bull Tensile mean stress accentuates the notch effect in fatigue hellip in fact Kf can exceed Kt in some cases Need to be catered for otherwise fatal in fatigue loading

bull Compressive mean stress can be beneficial It significantly reduces the effect of stress concentration and may in fact eliminate it

bull Residual stresses can contribute significantly to the level of mean stress This in turn may affect the notched behaviour

5192016

28

Questions

5192016

18

Goodman Diagram Constant Life Lines

Constructing failure envelopes using Goodman Diagram

Constant Life Lines

Constant Load Lines

httpneilwimerweeblycombasic-fatigue-analysishtml

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

5192016

19

Effect of R-ratio

Al Alloy 2024 T3Axial Loading

S max

ksi

N

102 103 104 105 106 107 108

40

60

0

80

20

73 ksi = UTS

R= SminSmax

+060

+050

+040

+025

+010

0

-030

-10

Ref Fatiue of Aircraft Structures NAVAIR 01-1A-13 1966

Questions

5192016

20

Numerical Example 1

In a Tension-Compression (R=-1) fatigue test with an alloy following mean fatigue lives were found for various stress levels

The ultimate strength of the alloy was found to be 462 MPa(i) Plot the Sa-N curve (ii) Check whether this obeys Basquin relation Find

Basquin exponent(iii) Using Goodman relation derive the Sa- N curve for

R=0

Smax (MPa) 138 172 207 276 345 380

N 12589250 1273500 199526 15850 1778 1060

Solution

0

100

200

300

400

10E+02 10E+04 10E+06 10E+08

Sa vs N

0

05

1

15

2

25

3

0 2 4 6 8

log Sa vs log N

Since R=-1 Sa=Smax

Basquin Exponent -0107Sfrsquo = 102892 = 780 MPa

WE take m=1 For R=0 Sm=Sa so that the eqn is 119878119886

1198781198860+

119878119898

119878119906

119898

= 1

119878119886

1198781198860+

119878119886

119878119906= 1rArr 119904119886 =

11

1198781198860+

1

119878119906

Substituting values of Sa0 we get Sa for each value of N Then we can plot

5192016

21

Solutionhellip continued

N 12589250 1273500 199526 15850 1778 1060

Smax (MPa) 138 172 207 276 345 380

Sa0 (R=-1) 138 172 207 276 345 380

Sa (R=0) 106 125 143 173 198 209

0

50

100

150

200

250

300

350

400

10E+02 10E+03 10E+04 10E+05 10E+06 10E+07 10E+08

Sa vs N

R = -1

R = 0

Numerical Example 2

For an Aluminium Alloy the S-N behaviour in tension compression fatigue with zero mean stress is approximated by the relation

119878119886= 480119873minus012

(i) What would be the life of a structure made of this alloy if it is subjected to cyclic stress of 80 MPa

(ii) What would be life if there is an additional steady stress of +10 MPa

(iii) What would be life if the additional steady stress in -10 MPa

5192016

22

Solution to Numerical Example 2

(i) This is direct substitution in the given stress-life curve equationWe have Sa = 80 MPa Therefore

80 = 480119873minus012

This gives N = (48080)^(1012) = 3052065

(ii) With an additional steady stress of +10 MPaHere the mean stress is now Sm=10 MPa and Sa=80 MPaSince mean stress is not zero we need to use Goodman relation to arrive at the mean stress effect

With mean stress zero 1198781198860= 119878119891prime119873119887

Recall Goodman relation

In absence data we take Su = Sfrsquo Now we can just substitute for Sa(=80) and Sm (=10) and Su=Sfrsquo = 480 with b= -012 and get N

Ans N= 2560937

(iii) Similarly with Sm = -10 we can repeat calculation

Ans 3624244

119878119886

1198781198860+

119878119898

119878119906= 1rArr

119878119886

119878119891prime119873119887 +

119878119898

119878119906= 1

Stress ConcentrationHole in a large plate under uniaxial tension

bull Large plate Max stress is 3 times the nominal stress (applied stress far-field stress)

bull Stress concentration Factor Kt = SmaxSn = 3

Elliptic hole SCF = 1+ 2(ba) = 1+2(a)

smax

sns

5192016

23

Stress Concentration (contd)

Petersonrsquos Original

compilationndash updated and

enhanced

5192016

24

Fatigue under Stress Concentration

S

N

s

s Krsquo (KrsquoltKt)

s Kt

Notched

Un-notched

Effective Stress Concentration for fatigue is less than the Stress Concentration in Static

Many types of discontinuities which act as stress concentrators Holes cut-outs key-slots cracks notches hellip

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119904119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ

119870119891 = 1 + 119902 119870119905 minus 1

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

Fatigue notch sensitivity factor

Stress Concentration Factor

119870119905 =119872119886119909 119904119905119903119890119904119904 119908119894119905ℎ 119899119900119905119888ℎ

119878119905119903119890119904119904 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ 1 lt 119870119891 lt 119870119905

Fatigue Notch Sensitivity Factor q119870119891 = 1 + 119902 119870119905 minus 1 and

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

5192016

25

Estimates of 119902 ∶ All Empirical

bull Neuberrsquos Approx Formula (R=-1) 119954 =120783

120783+ Τ120646 119955(r = notch root radius 120588 = material characteristic length)bull For aluminium alloys

Ult Strength 150 300 600 MPa120588 20 06 04 mm

bull Petersonrsquos Approx Formula (R=-1) 119954 =120783

120783+119938

119955(r = notch root radius 119886 = another mat characteristic length)

bull For Steels 119886 = 002542070

119878119906

18

(Su in MPa a in mm)

bull For Al-alloys 119886 = 0635 119898119898 119886119901119901119903119900119909

NeuberParameter

0

005

01

015

02

025

03

0 500 1000 1500

Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500 1000 1500

Ne

ub

er

Par

ame

ter

Su MPa

Al-Alloys

Steels

5192016

26

Goodman Diagram for notched fatigue

bull Rather simplisticbull Does not

account for variation of Kf

with mean stress or life cycles

S a

Sa

Accounting for means stress effectHaig Diagrams

5192016

27

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

More about notched fatiguehellip

bull Mean stress effect is more pronounced in notched specimens than in smooth ones

bull Tensile mean stress accentuates the notch effect in fatigue hellip in fact Kf can exceed Kt in some cases Need to be catered for otherwise fatal in fatigue loading

bull Compressive mean stress can be beneficial It significantly reduces the effect of stress concentration and may in fact eliminate it

bull Residual stresses can contribute significantly to the level of mean stress This in turn may affect the notched behaviour

5192016

28

Questions

5192016

19

Effect of R-ratio

Al Alloy 2024 T3Axial Loading

S max

ksi

N

102 103 104 105 106 107 108

40

60

0

80

20

73 ksi = UTS

R= SminSmax

+060

+050

+040

+025

+010

0

-030

-10

Ref Fatiue of Aircraft Structures NAVAIR 01-1A-13 1966

Questions

5192016

20

Numerical Example 1

In a Tension-Compression (R=-1) fatigue test with an alloy following mean fatigue lives were found for various stress levels

The ultimate strength of the alloy was found to be 462 MPa(i) Plot the Sa-N curve (ii) Check whether this obeys Basquin relation Find

Basquin exponent(iii) Using Goodman relation derive the Sa- N curve for

R=0

Smax (MPa) 138 172 207 276 345 380

N 12589250 1273500 199526 15850 1778 1060

Solution

0

100

200

300

400

10E+02 10E+04 10E+06 10E+08

Sa vs N

0

05

1

15

2

25

3

0 2 4 6 8

log Sa vs log N

Since R=-1 Sa=Smax

Basquin Exponent -0107Sfrsquo = 102892 = 780 MPa

WE take m=1 For R=0 Sm=Sa so that the eqn is 119878119886

1198781198860+

119878119898

119878119906

119898

= 1

119878119886

1198781198860+

119878119886

119878119906= 1rArr 119904119886 =

11

1198781198860+

1

119878119906

Substituting values of Sa0 we get Sa for each value of N Then we can plot

5192016

21

Solutionhellip continued

N 12589250 1273500 199526 15850 1778 1060

Smax (MPa) 138 172 207 276 345 380

Sa0 (R=-1) 138 172 207 276 345 380

Sa (R=0) 106 125 143 173 198 209

0

50

100

150

200

250

300

350

400

10E+02 10E+03 10E+04 10E+05 10E+06 10E+07 10E+08

Sa vs N

R = -1

R = 0

Numerical Example 2

For an Aluminium Alloy the S-N behaviour in tension compression fatigue with zero mean stress is approximated by the relation

119878119886= 480119873minus012

(i) What would be the life of a structure made of this alloy if it is subjected to cyclic stress of 80 MPa

(ii) What would be life if there is an additional steady stress of +10 MPa

(iii) What would be life if the additional steady stress in -10 MPa

5192016

22

Solution to Numerical Example 2

(i) This is direct substitution in the given stress-life curve equationWe have Sa = 80 MPa Therefore

80 = 480119873minus012

This gives N = (48080)^(1012) = 3052065

(ii) With an additional steady stress of +10 MPaHere the mean stress is now Sm=10 MPa and Sa=80 MPaSince mean stress is not zero we need to use Goodman relation to arrive at the mean stress effect

With mean stress zero 1198781198860= 119878119891prime119873119887

Recall Goodman relation

In absence data we take Su = Sfrsquo Now we can just substitute for Sa(=80) and Sm (=10) and Su=Sfrsquo = 480 with b= -012 and get N

Ans N= 2560937

(iii) Similarly with Sm = -10 we can repeat calculation

Ans 3624244

119878119886

1198781198860+

119878119898

119878119906= 1rArr

119878119886

119878119891prime119873119887 +

119878119898

119878119906= 1

Stress ConcentrationHole in a large plate under uniaxial tension

bull Large plate Max stress is 3 times the nominal stress (applied stress far-field stress)

bull Stress concentration Factor Kt = SmaxSn = 3

Elliptic hole SCF = 1+ 2(ba) = 1+2(a)

smax

sns

5192016

23

Stress Concentration (contd)

Petersonrsquos Original

compilationndash updated and

enhanced

5192016

24

Fatigue under Stress Concentration

S

N

s

s Krsquo (KrsquoltKt)

s Kt

Notched

Un-notched

Effective Stress Concentration for fatigue is less than the Stress Concentration in Static

Many types of discontinuities which act as stress concentrators Holes cut-outs key-slots cracks notches hellip

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119904119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ

119870119891 = 1 + 119902 119870119905 minus 1

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

Fatigue notch sensitivity factor

Stress Concentration Factor

119870119905 =119872119886119909 119904119905119903119890119904119904 119908119894119905ℎ 119899119900119905119888ℎ

119878119905119903119890119904119904 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ 1 lt 119870119891 lt 119870119905

Fatigue Notch Sensitivity Factor q119870119891 = 1 + 119902 119870119905 minus 1 and

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

5192016

25

Estimates of 119902 ∶ All Empirical

bull Neuberrsquos Approx Formula (R=-1) 119954 =120783

120783+ Τ120646 119955(r = notch root radius 120588 = material characteristic length)bull For aluminium alloys

Ult Strength 150 300 600 MPa120588 20 06 04 mm

bull Petersonrsquos Approx Formula (R=-1) 119954 =120783

120783+119938

119955(r = notch root radius 119886 = another mat characteristic length)

bull For Steels 119886 = 002542070

119878119906

18

(Su in MPa a in mm)

bull For Al-alloys 119886 = 0635 119898119898 119886119901119901119903119900119909

NeuberParameter

0

005

01

015

02

025

03

0 500 1000 1500

Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500 1000 1500

Ne

ub

er

Par

ame

ter

Su MPa

Al-Alloys

Steels

5192016

26

Goodman Diagram for notched fatigue

bull Rather simplisticbull Does not

account for variation of Kf

with mean stress or life cycles

S a

Sa

Accounting for means stress effectHaig Diagrams

5192016

27

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

More about notched fatiguehellip

bull Mean stress effect is more pronounced in notched specimens than in smooth ones

bull Tensile mean stress accentuates the notch effect in fatigue hellip in fact Kf can exceed Kt in some cases Need to be catered for otherwise fatal in fatigue loading

bull Compressive mean stress can be beneficial It significantly reduces the effect of stress concentration and may in fact eliminate it

bull Residual stresses can contribute significantly to the level of mean stress This in turn may affect the notched behaviour

5192016

28

Questions

5192016

20

Numerical Example 1

In a Tension-Compression (R=-1) fatigue test with an alloy following mean fatigue lives were found for various stress levels

The ultimate strength of the alloy was found to be 462 MPa(i) Plot the Sa-N curve (ii) Check whether this obeys Basquin relation Find

Basquin exponent(iii) Using Goodman relation derive the Sa- N curve for

R=0

Smax (MPa) 138 172 207 276 345 380

N 12589250 1273500 199526 15850 1778 1060

Solution

0

100

200

300

400

10E+02 10E+04 10E+06 10E+08

Sa vs N

0

05

1

15

2

25

3

0 2 4 6 8

log Sa vs log N

Since R=-1 Sa=Smax

Basquin Exponent -0107Sfrsquo = 102892 = 780 MPa

WE take m=1 For R=0 Sm=Sa so that the eqn is 119878119886

1198781198860+

119878119898

119878119906

119898

= 1

119878119886

1198781198860+

119878119886

119878119906= 1rArr 119904119886 =

11

1198781198860+

1

119878119906

Substituting values of Sa0 we get Sa for each value of N Then we can plot

5192016

21

Solutionhellip continued

N 12589250 1273500 199526 15850 1778 1060

Smax (MPa) 138 172 207 276 345 380

Sa0 (R=-1) 138 172 207 276 345 380

Sa (R=0) 106 125 143 173 198 209

0

50

100

150

200

250

300

350

400

10E+02 10E+03 10E+04 10E+05 10E+06 10E+07 10E+08

Sa vs N

R = -1

R = 0

Numerical Example 2

For an Aluminium Alloy the S-N behaviour in tension compression fatigue with zero mean stress is approximated by the relation

119878119886= 480119873minus012

(i) What would be the life of a structure made of this alloy if it is subjected to cyclic stress of 80 MPa

(ii) What would be life if there is an additional steady stress of +10 MPa

(iii) What would be life if the additional steady stress in -10 MPa

5192016

22

Solution to Numerical Example 2

(i) This is direct substitution in the given stress-life curve equationWe have Sa = 80 MPa Therefore

80 = 480119873minus012

This gives N = (48080)^(1012) = 3052065

(ii) With an additional steady stress of +10 MPaHere the mean stress is now Sm=10 MPa and Sa=80 MPaSince mean stress is not zero we need to use Goodman relation to arrive at the mean stress effect

With mean stress zero 1198781198860= 119878119891prime119873119887

Recall Goodman relation

In absence data we take Su = Sfrsquo Now we can just substitute for Sa(=80) and Sm (=10) and Su=Sfrsquo = 480 with b= -012 and get N

Ans N= 2560937

(iii) Similarly with Sm = -10 we can repeat calculation

Ans 3624244

119878119886

1198781198860+

119878119898

119878119906= 1rArr

119878119886

119878119891prime119873119887 +

119878119898

119878119906= 1

Stress ConcentrationHole in a large plate under uniaxial tension

bull Large plate Max stress is 3 times the nominal stress (applied stress far-field stress)

bull Stress concentration Factor Kt = SmaxSn = 3

Elliptic hole SCF = 1+ 2(ba) = 1+2(a)

smax

sns

5192016

23

Stress Concentration (contd)

Petersonrsquos Original

compilationndash updated and

enhanced

5192016

24

Fatigue under Stress Concentration

S

N

s

s Krsquo (KrsquoltKt)

s Kt

Notched

Un-notched

Effective Stress Concentration for fatigue is less than the Stress Concentration in Static

Many types of discontinuities which act as stress concentrators Holes cut-outs key-slots cracks notches hellip

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119904119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ

119870119891 = 1 + 119902 119870119905 minus 1

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

Fatigue notch sensitivity factor

Stress Concentration Factor

119870119905 =119872119886119909 119904119905119903119890119904119904 119908119894119905ℎ 119899119900119905119888ℎ

119878119905119903119890119904119904 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ 1 lt 119870119891 lt 119870119905

Fatigue Notch Sensitivity Factor q119870119891 = 1 + 119902 119870119905 minus 1 and

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

5192016

25

Estimates of 119902 ∶ All Empirical

bull Neuberrsquos Approx Formula (R=-1) 119954 =120783

120783+ Τ120646 119955(r = notch root radius 120588 = material characteristic length)bull For aluminium alloys

Ult Strength 150 300 600 MPa120588 20 06 04 mm

bull Petersonrsquos Approx Formula (R=-1) 119954 =120783

120783+119938

119955(r = notch root radius 119886 = another mat characteristic length)

bull For Steels 119886 = 002542070

119878119906

18

(Su in MPa a in mm)

bull For Al-alloys 119886 = 0635 119898119898 119886119901119901119903119900119909

NeuberParameter

0

005

01

015

02

025

03

0 500 1000 1500

Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500 1000 1500

Ne

ub

er

Par

ame

ter

Su MPa

Al-Alloys

Steels

5192016

26

Goodman Diagram for notched fatigue

bull Rather simplisticbull Does not

account for variation of Kf

with mean stress or life cycles

S a

Sa

Accounting for means stress effectHaig Diagrams

5192016

27

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

More about notched fatiguehellip

bull Mean stress effect is more pronounced in notched specimens than in smooth ones

bull Tensile mean stress accentuates the notch effect in fatigue hellip in fact Kf can exceed Kt in some cases Need to be catered for otherwise fatal in fatigue loading

bull Compressive mean stress can be beneficial It significantly reduces the effect of stress concentration and may in fact eliminate it

bull Residual stresses can contribute significantly to the level of mean stress This in turn may affect the notched behaviour

5192016

28

Questions

5192016

21

Solutionhellip continued

N 12589250 1273500 199526 15850 1778 1060

Smax (MPa) 138 172 207 276 345 380

Sa0 (R=-1) 138 172 207 276 345 380

Sa (R=0) 106 125 143 173 198 209

0

50

100

150

200

250

300

350

400

10E+02 10E+03 10E+04 10E+05 10E+06 10E+07 10E+08

Sa vs N

R = -1

R = 0

Numerical Example 2

For an Aluminium Alloy the S-N behaviour in tension compression fatigue with zero mean stress is approximated by the relation

119878119886= 480119873minus012

(i) What would be the life of a structure made of this alloy if it is subjected to cyclic stress of 80 MPa

(ii) What would be life if there is an additional steady stress of +10 MPa

(iii) What would be life if the additional steady stress in -10 MPa

5192016

22

Solution to Numerical Example 2

(i) This is direct substitution in the given stress-life curve equationWe have Sa = 80 MPa Therefore

80 = 480119873minus012

This gives N = (48080)^(1012) = 3052065

(ii) With an additional steady stress of +10 MPaHere the mean stress is now Sm=10 MPa and Sa=80 MPaSince mean stress is not zero we need to use Goodman relation to arrive at the mean stress effect

With mean stress zero 1198781198860= 119878119891prime119873119887

Recall Goodman relation

In absence data we take Su = Sfrsquo Now we can just substitute for Sa(=80) and Sm (=10) and Su=Sfrsquo = 480 with b= -012 and get N

Ans N= 2560937

(iii) Similarly with Sm = -10 we can repeat calculation

Ans 3624244

119878119886

1198781198860+

119878119898

119878119906= 1rArr

119878119886

119878119891prime119873119887 +

119878119898

119878119906= 1

Stress ConcentrationHole in a large plate under uniaxial tension

bull Large plate Max stress is 3 times the nominal stress (applied stress far-field stress)

bull Stress concentration Factor Kt = SmaxSn = 3

Elliptic hole SCF = 1+ 2(ba) = 1+2(a)

smax

sns

5192016

23

Stress Concentration (contd)

Petersonrsquos Original

compilationndash updated and

enhanced

5192016

24

Fatigue under Stress Concentration

S

N

s

s Krsquo (KrsquoltKt)

s Kt

Notched

Un-notched

Effective Stress Concentration for fatigue is less than the Stress Concentration in Static

Many types of discontinuities which act as stress concentrators Holes cut-outs key-slots cracks notches hellip

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119904119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ

119870119891 = 1 + 119902 119870119905 minus 1

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

Fatigue notch sensitivity factor

Stress Concentration Factor

119870119905 =119872119886119909 119904119905119903119890119904119904 119908119894119905ℎ 119899119900119905119888ℎ

119878119905119903119890119904119904 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ 1 lt 119870119891 lt 119870119905

Fatigue Notch Sensitivity Factor q119870119891 = 1 + 119902 119870119905 minus 1 and

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

5192016

25

Estimates of 119902 ∶ All Empirical

bull Neuberrsquos Approx Formula (R=-1) 119954 =120783

120783+ Τ120646 119955(r = notch root radius 120588 = material characteristic length)bull For aluminium alloys

Ult Strength 150 300 600 MPa120588 20 06 04 mm

bull Petersonrsquos Approx Formula (R=-1) 119954 =120783

120783+119938

119955(r = notch root radius 119886 = another mat characteristic length)

bull For Steels 119886 = 002542070

119878119906

18

(Su in MPa a in mm)

bull For Al-alloys 119886 = 0635 119898119898 119886119901119901119903119900119909

NeuberParameter

0

005

01

015

02

025

03

0 500 1000 1500

Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500 1000 1500

Ne

ub

er

Par

ame

ter

Su MPa

Al-Alloys

Steels

5192016

26

Goodman Diagram for notched fatigue

bull Rather simplisticbull Does not

account for variation of Kf

with mean stress or life cycles

S a

Sa

Accounting for means stress effectHaig Diagrams

5192016

27

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

More about notched fatiguehellip

bull Mean stress effect is more pronounced in notched specimens than in smooth ones

bull Tensile mean stress accentuates the notch effect in fatigue hellip in fact Kf can exceed Kt in some cases Need to be catered for otherwise fatal in fatigue loading

bull Compressive mean stress can be beneficial It significantly reduces the effect of stress concentration and may in fact eliminate it

bull Residual stresses can contribute significantly to the level of mean stress This in turn may affect the notched behaviour

5192016

28

Questions

5192016

22

Solution to Numerical Example 2

(i) This is direct substitution in the given stress-life curve equationWe have Sa = 80 MPa Therefore

80 = 480119873minus012

This gives N = (48080)^(1012) = 3052065

(ii) With an additional steady stress of +10 MPaHere the mean stress is now Sm=10 MPa and Sa=80 MPaSince mean stress is not zero we need to use Goodman relation to arrive at the mean stress effect

With mean stress zero 1198781198860= 119878119891prime119873119887

Recall Goodman relation

In absence data we take Su = Sfrsquo Now we can just substitute for Sa(=80) and Sm (=10) and Su=Sfrsquo = 480 with b= -012 and get N

Ans N= 2560937

(iii) Similarly with Sm = -10 we can repeat calculation

Ans 3624244

119878119886

1198781198860+

119878119898

119878119906= 1rArr

119878119886

119878119891prime119873119887 +

119878119898

119878119906= 1

Stress ConcentrationHole in a large plate under uniaxial tension

bull Large plate Max stress is 3 times the nominal stress (applied stress far-field stress)

bull Stress concentration Factor Kt = SmaxSn = 3

Elliptic hole SCF = 1+ 2(ba) = 1+2(a)

smax

sns

5192016

23

Stress Concentration (contd)

Petersonrsquos Original

compilationndash updated and

enhanced

5192016

24

Fatigue under Stress Concentration

S

N

s

s Krsquo (KrsquoltKt)

s Kt

Notched

Un-notched

Effective Stress Concentration for fatigue is less than the Stress Concentration in Static

Many types of discontinuities which act as stress concentrators Holes cut-outs key-slots cracks notches hellip

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119904119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ

119870119891 = 1 + 119902 119870119905 minus 1

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

Fatigue notch sensitivity factor

Stress Concentration Factor

119870119905 =119872119886119909 119904119905119903119890119904119904 119908119894119905ℎ 119899119900119905119888ℎ

119878119905119903119890119904119904 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ 1 lt 119870119891 lt 119870119905

Fatigue Notch Sensitivity Factor q119870119891 = 1 + 119902 119870119905 minus 1 and

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

5192016

25

Estimates of 119902 ∶ All Empirical

bull Neuberrsquos Approx Formula (R=-1) 119954 =120783

120783+ Τ120646 119955(r = notch root radius 120588 = material characteristic length)bull For aluminium alloys

Ult Strength 150 300 600 MPa120588 20 06 04 mm

bull Petersonrsquos Approx Formula (R=-1) 119954 =120783

120783+119938

119955(r = notch root radius 119886 = another mat characteristic length)

bull For Steels 119886 = 002542070

119878119906

18

(Su in MPa a in mm)

bull For Al-alloys 119886 = 0635 119898119898 119886119901119901119903119900119909

NeuberParameter

0

005

01

015

02

025

03

0 500 1000 1500

Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500 1000 1500

Ne

ub

er

Par

ame

ter

Su MPa

Al-Alloys

Steels

5192016

26

Goodman Diagram for notched fatigue

bull Rather simplisticbull Does not

account for variation of Kf

with mean stress or life cycles

S a

Sa

Accounting for means stress effectHaig Diagrams

5192016

27

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

More about notched fatiguehellip

bull Mean stress effect is more pronounced in notched specimens than in smooth ones

bull Tensile mean stress accentuates the notch effect in fatigue hellip in fact Kf can exceed Kt in some cases Need to be catered for otherwise fatal in fatigue loading

bull Compressive mean stress can be beneficial It significantly reduces the effect of stress concentration and may in fact eliminate it

bull Residual stresses can contribute significantly to the level of mean stress This in turn may affect the notched behaviour

5192016

28

Questions

5192016

23

Stress Concentration (contd)

Petersonrsquos Original

compilationndash updated and

enhanced

5192016

24

Fatigue under Stress Concentration

S

N

s

s Krsquo (KrsquoltKt)

s Kt

Notched

Un-notched

Effective Stress Concentration for fatigue is less than the Stress Concentration in Static

Many types of discontinuities which act as stress concentrators Holes cut-outs key-slots cracks notches hellip

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119904119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ

119870119891 = 1 + 119902 119870119905 minus 1

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

Fatigue notch sensitivity factor

Stress Concentration Factor

119870119905 =119872119886119909 119904119905119903119890119904119904 119908119894119905ℎ 119899119900119905119888ℎ

119878119905119903119890119904119904 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ 1 lt 119870119891 lt 119870119905

Fatigue Notch Sensitivity Factor q119870119891 = 1 + 119902 119870119905 minus 1 and

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

5192016

25

Estimates of 119902 ∶ All Empirical

bull Neuberrsquos Approx Formula (R=-1) 119954 =120783

120783+ Τ120646 119955(r = notch root radius 120588 = material characteristic length)bull For aluminium alloys

Ult Strength 150 300 600 MPa120588 20 06 04 mm

bull Petersonrsquos Approx Formula (R=-1) 119954 =120783

120783+119938

119955(r = notch root radius 119886 = another mat characteristic length)

bull For Steels 119886 = 002542070

119878119906

18

(Su in MPa a in mm)

bull For Al-alloys 119886 = 0635 119898119898 119886119901119901119903119900119909

NeuberParameter

0

005

01

015

02

025

03

0 500 1000 1500

Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500 1000 1500

Ne

ub

er

Par

ame

ter

Su MPa

Al-Alloys

Steels

5192016

26

Goodman Diagram for notched fatigue

bull Rather simplisticbull Does not

account for variation of Kf

with mean stress or life cycles

S a

Sa

Accounting for means stress effectHaig Diagrams

5192016

27

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

More about notched fatiguehellip

bull Mean stress effect is more pronounced in notched specimens than in smooth ones

bull Tensile mean stress accentuates the notch effect in fatigue hellip in fact Kf can exceed Kt in some cases Need to be catered for otherwise fatal in fatigue loading

bull Compressive mean stress can be beneficial It significantly reduces the effect of stress concentration and may in fact eliminate it

bull Residual stresses can contribute significantly to the level of mean stress This in turn may affect the notched behaviour

5192016

28

Questions

5192016

24

Fatigue under Stress Concentration

S

N

s

s Krsquo (KrsquoltKt)

s Kt

Notched

Un-notched

Effective Stress Concentration for fatigue is less than the Stress Concentration in Static

Many types of discontinuities which act as stress concentrators Holes cut-outs key-slots cracks notches hellip

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119904119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ

119870119891 = 1 + 119902 119870119905 minus 1

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

Fatigue notch sensitivity factor

Stress Concentration Factor

119870119905 =119872119886119909 119904119905119903119890119904119904 119908119894119905ℎ 119899119900119905119888ℎ

119878119905119903119890119904119904 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

Fatigue Concentration Factor

119870119891 =119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ119900119906119905 119899119900119905119888ℎ

119865119886119905119894119892119906119890 119878119905119903119890119899119892119905ℎ 119908119894119905ℎ 119899119900119905119888ℎ 1 lt 119870119891 lt 119870119905

Fatigue Notch Sensitivity Factor q119870119891 = 1 + 119902 119870119905 minus 1 and

119902 =119870119891 minus 1

119870119905 minus 1 0 lt 119902 lt 1

5192016

25

Estimates of 119902 ∶ All Empirical

bull Neuberrsquos Approx Formula (R=-1) 119954 =120783

120783+ Τ120646 119955(r = notch root radius 120588 = material characteristic length)bull For aluminium alloys

Ult Strength 150 300 600 MPa120588 20 06 04 mm

bull Petersonrsquos Approx Formula (R=-1) 119954 =120783

120783+119938

119955(r = notch root radius 119886 = another mat characteristic length)

bull For Steels 119886 = 002542070

119878119906

18

(Su in MPa a in mm)

bull For Al-alloys 119886 = 0635 119898119898 119886119901119901119903119900119909

NeuberParameter

0

005

01

015

02

025

03

0 500 1000 1500

Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500 1000 1500

Ne

ub

er

Par

ame

ter

Su MPa

Al-Alloys

Steels

5192016

26

Goodman Diagram for notched fatigue

bull Rather simplisticbull Does not

account for variation of Kf

with mean stress or life cycles

S a

Sa

Accounting for means stress effectHaig Diagrams

5192016

27

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

More about notched fatiguehellip

bull Mean stress effect is more pronounced in notched specimens than in smooth ones

bull Tensile mean stress accentuates the notch effect in fatigue hellip in fact Kf can exceed Kt in some cases Need to be catered for otherwise fatal in fatigue loading

bull Compressive mean stress can be beneficial It significantly reduces the effect of stress concentration and may in fact eliminate it

bull Residual stresses can contribute significantly to the level of mean stress This in turn may affect the notched behaviour

5192016

28

Questions

5192016

25

Estimates of 119902 ∶ All Empirical

bull Neuberrsquos Approx Formula (R=-1) 119954 =120783

120783+ Τ120646 119955(r = notch root radius 120588 = material characteristic length)bull For aluminium alloys

Ult Strength 150 300 600 MPa120588 20 06 04 mm

bull Petersonrsquos Approx Formula (R=-1) 119954 =120783

120783+119938

119955(r = notch root radius 119886 = another mat characteristic length)

bull For Steels 119886 = 002542070

119878119906

18

(Su in MPa a in mm)

bull For Al-alloys 119886 = 0635 119898119898 119886119901119901119903119900119909

NeuberParameter

0

005

01

015

02

025

03

0 500 1000 1500

Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500Ne

ub

er

Par

amet

er

(

mm

)

Su MPa

0

05

1

15

2

25

0 500 1000 1500

Ne

ub

er

Par

ame

ter

Su MPa

Al-Alloys

Steels

5192016

26

Goodman Diagram for notched fatigue

bull Rather simplisticbull Does not

account for variation of Kf

with mean stress or life cycles

S a

Sa

Accounting for means stress effectHaig Diagrams

5192016

27

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

More about notched fatiguehellip

bull Mean stress effect is more pronounced in notched specimens than in smooth ones

bull Tensile mean stress accentuates the notch effect in fatigue hellip in fact Kf can exceed Kt in some cases Need to be catered for otherwise fatal in fatigue loading

bull Compressive mean stress can be beneficial It significantly reduces the effect of stress concentration and may in fact eliminate it

bull Residual stresses can contribute significantly to the level of mean stress This in turn may affect the notched behaviour

5192016

28

Questions

5192016

26

Goodman Diagram for notched fatigue

bull Rather simplisticbull Does not

account for variation of Kf

with mean stress or life cycles

S a

Sa

Accounting for means stress effectHaig Diagrams

5192016

27

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

More about notched fatiguehellip

bull Mean stress effect is more pronounced in notched specimens than in smooth ones

bull Tensile mean stress accentuates the notch effect in fatigue hellip in fact Kf can exceed Kt in some cases Need to be catered for otherwise fatal in fatigue loading

bull Compressive mean stress can be beneficial It significantly reduces the effect of stress concentration and may in fact eliminate it

bull Residual stresses can contribute significantly to the level of mean stress This in turn may affect the notched behaviour

5192016

28

Questions

5192016

27

WADC TECHNICAL REPORT 52-307 PART 1 FATIGUE PROPERTIES OF ALUMINUM ALLOYS AT VARIOUS DIRECT STRESS RATIOS ADS0007610 1955

A= SaSm

More about notched fatiguehellip

bull Mean stress effect is more pronounced in notched specimens than in smooth ones

bull Tensile mean stress accentuates the notch effect in fatigue hellip in fact Kf can exceed Kt in some cases Need to be catered for otherwise fatal in fatigue loading

bull Compressive mean stress can be beneficial It significantly reduces the effect of stress concentration and may in fact eliminate it

bull Residual stresses can contribute significantly to the level of mean stress This in turn may affect the notched behaviour

5192016

28

Questions

5192016

28

Questions