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Concentration factors

y

x

S

SK t

0

1

x

y

xG

Shape Factor orStress Concentration Factor ofan Elastic Stress

Relative Stress Gradient

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http://mechanika.fs.cvut.cz/calculator.php

Concentration factors

15

2

R 4

20

10

111

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Concentration factors

https://www.efatigue.com/constantamplitude/stressconcentration/

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Fatigue notch factor

0

100

200

300

400

500

600

1,E+03 1,E+04 1,E+05 1,E+06 1,E+07 1,E+08

Number of cycles [1]

Str

es

s a

mp

litu

de

[M

Pa

]

smooth

notched

FL

FL,N

,

FLf

FL N

K

1t fK K q

Theoretically, if material has high notch sensitivity q

Fatigue notch factor

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Fatigue notch factor, Kf and notch sensitivity factor, q

Experiments have shown that the effect of notches is less than that the estimated effect according a traditional stress concentration factor, Kt.

The fatigue notch factor, Kf , can be described as the effective stress concentration in fatigue.

This effect is dealt with using a notch sensitivity factor, q.

1 1f tK K q

Notch radius

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Fatigue limit modifiing factors

Loading factor, kL

Historically, fatigue limits have been determined from simple bending tests where there is a stress gradient in the test specimen. A specimen loaded in tension will have a lower fatigue limit than one loaded in bending. An empirical correction factor, called the loading factor, is used to make an allowance for this effect.

Loading Type kL

Axial 0.9Bending 1.0Torsion 0.57

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Fatigue limit modifiing factors

Surface finish factor, kSF

Fatigue limits are determined from small polished laboratory specimens. A surface finish correction is made to estimate the fatigue limit of the part with the actual surface finish

kSF

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Fatigue limit modifiing factors

Size factor, kS

Experimentally, larger parts have lower fatigue limits than smaller parts. Since the materials data is obtained from small specimens, a correction factor, called the size factor, is used for larger diameters.

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Fatigue limit modifiing factors

Size factor, kS

For non-circular sections an effective diameter is computed. The effective diameter is obtained by equating the volume of material subjected to 95% of the maximum stress to a round bar in bending with the same highly stressed volume

exp

10exp

mDDFL

S d dFL

Vk

V

y

x

S

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Fatigue limit modifiing factors

Surface treatment factor, kT

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Fatigue limit of a real part

Estimation of the fatigue limit of a real part

,FL L SF S T

FL Nf

k k k k

K

1. CTU in Prague, Faculty of Mechanical Engineering 1. DAF 1. 1. Page 12

Safety factor of unlimited fatigue life (permanent strength)

1. Alternating stress (R=-1)• operational loading stress amplitude a

• fatigue limit of the real part in the critical cross section area FL,N

a

NFLn

,

100

1000

1.E+04 1.E+05 1.E+06 1.E+07

N [1]

a [

MP

a]

alloy steel

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Example – Fatigue safe factor calculation

Railway axle

Material: alloy steel 24CrMo4, ASTM 4130

Point A of the potential crack initiation

Experimental strain amplitude measurement (in the point A):

Problem description:

,max 312 microstraina

A

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Example – continuation http://fatiguecalculator.com

0,0762 1570 2

b

a f N N

0,07671570 2 10 437.5 MPaFL

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Example – continuation http://fatiguecalculator.com

2.09tK

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Example – continuation FEM Calculation – CTU Prague

1.95tK

Wheel

Braking disc

Axle

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Example – continuation

1.95tK

1 1 1 2.0 1 0.83 1.83f tK K q

15

0

100

200

300

400

500

600

1,E+03 1,E+04 1,E+05 1,E+06 1,E+07 1,E+08

Number of cycles [1]

Str

es

s a

mp

litu

de

[M

Pa

]

smooth

notched

FL

FL,N

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Fatigue Limit

Estimation of the fatigue limit of a real part

,FL L SF S T

FL Nf

k k k k

K

factor k value

loading kL 1.00

surface finish kSF 0.67

size factor kS 0.70

size factor kT 1.00

,

437.5 1.00 0.67 0.70 1.00112.1 MPa

1.83FL N

Example – continuation

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Estimation of the nominal stress amplitude

Experimental strain amplitude measurement (in the point A):

,max 512 microstraina

A

206850 0.000312 64.5 MPaa E

Fatigue loadingExample – continuation

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Safety factor

Estimation of the safety factor nFL

A 64.5 MPaa

Example – continuation

, 112.1 MPaFL N

0

100

200

300

400

500

600

1,E+03 1,E+04 1,E+05 1,E+06 1,E+07 1,E+08

Number of cycles [1]

Str

es

s a

mp

litu

de

[M

Pa

]

smooth

notched

FL

FL,N

, 112.11.74

64.5FL N

FLa

n

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Questions and problems II

.

1. What is difference between the shape factor and the notch factor? Write their definition equations.

2. Define the notch sensitivity factor of material and write equation for it (as a function of shape and notch factor).

3. Depends the stress concentration factor of metals on a material parameters? And what about of the notch factor?

4. What is the typical value of the stress concentration factor at a large tensile loaded plate with a round hole in the middle? Is the notch factor of such plate lower or higher as the shape factor?

5. Is the fatigue limit of a real part the same as the fatigue limit of a basic material? What other factors could be taken in the account by an expression of such fatigue limit?

6. What dimension of a shaft has higher the size factor ks ? Shaft with higher or smaller diameter?

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Questions and problems II

.

Estimate the fatigue limit and the safety factor of the part on the Fig. 1. under harmonic loading.

The material of the part is steel 4130 bar Su= 778 MPasee http://fatiguecalculator.com/cgi-bin/StressShowMatProp.pl

Fig. 1

Other inputs:D= 50 mm outer diameterd= 30 mm inner diameterr= 5 mm notch radius

F= 40 kN maximum loading forceq= 0,8 1 notch sensitivity of the steelR= -1 stress ratio

n L= ? fatigue life safety factorN FL 2,00E+06 number of fatigue limit cyclesRa= 12,5 surface roughness

Example:

Solution

shape factor Kt= 1,71notch factor Kf= 1,57fatigue limit sFL= 7,95E+01safety factor nF= 1,40E+00