chapter05(design against for fluctuating load)

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Chapter 5

Design against Fluctuating

Load

授課教師:尤春風

I

yM bb

J

rM tA

Pt

Elementary equations:

Stress concentration is defined as the localization of high stresses due

to the irregularities presents in the component and abrupt changes of

the cross section.

Stress concentration factor ( ) is defined as

section -cross minimalfor equations elementaryby obtained stresses nominal

itydiscontinunear stress actual of luehighest vatK

5.1 Stress concentration

tK

Fig. 5.1 Stress Concentration

Fig. 5.2 Stress Concentration Factor (Rectangular Plate with

Transverse Hole in Tension or Compression)

5.2 Stress concentration factor

The nominal stress is given by

where t is the plate thickness.

tdw

P

)(0

Fig. 5.3 Stress Concentration Factor (Flat Plate with Shoulder Fillet in

Tension or Compression)

td

P

0

The nominal stress is given by

where t is the plate thickness.

Fig. 5.4 Stress Concentration Factor (Round haft with Shoulder

Fillet in Tension)

20

4d

P

The nominal stress is given by

where d is the diameter on the

small end.

Fig. 5.5 Stress Concentration Factor (Round Shaft with Shoulder

Fillet in Bending)

I

yM b 0

The nominal stress is given by

where d is the diameter on the

smaller end.

2 and

64

4 dy

dI

Fig. 5.6 Stress Concentration Factor (Round Shaft with Fillet in Torsion)

The nominal stress is given by

where d is the diameter on the

smaller end.

2 and

32

4 dr

dJ

J

rM t 0

Fig. 5.7 Stress Concentration due to Elliptical Hole

)(21b

aKt

a = half width (or semi-axis) of ellipse perpendicular to the direction

of load

b = half width (or semi-axis) of ellipse in the direction of load

Following guidelines are considered for the stress

concentration factor:

(1)Ductile material under the static load

When the stress in the vicinity of the discontinuity

reaches the yield point, there is plastic deformation,

resulting in a redistribution of stresses. This plastic

deformation or yielding is local and restricted to very

small area in the component. There is no perceptible

damage to the part as a whole.

(2) Ductile material under the fluctuating load

When the load is fluctuating, the stresses at the

discontinuity exceed the endurance limit, the

component may fail. Therefore, endurance limit of the

components made of the ductile material is greatly

reduced due to stress concentration.

Following guidelines are considered for the stress

concentration factor:

(3) Brittle material

The effect of stress concentration is more severe in case of

brittle material, due to their instability to plastic

deformation.

5.3 Reduction of stress concentation

Fluid mechanics Solid mechanics

volume flow force

velocity stress

Flow pattern intensity Stress concentration factor

dAF dAuq

Flow analogy

Fig. 5.8 Force Flow Analogy

(a) Force Flow around Sharp Corner

(b) Force Flow around Rounded Corner

Fig. 5.9 Reduction of Stress Concentration due to V-notch

(a) Original Notch (b) Multiple Notches

(c) Drilled Holes (d) Removal of Undesirable Material

Fig. 5.10 Reduction of Stress Concentration due to Abrupt

Change in Cross-section

(a) Original Component (b) Fillet Radius

(C) Undercutting (d) Addition of Notch

Fig. 5.11 Reduction of Stress Concentration in Shaft with Keyway

(a) Original Shaft (b) Drilled Holes (c) Fillet Radius

Fig. 5.12 Reduction of Stress Concentration in Threaded Components

(a) Original Component (b) Undercutting

(c) Reduction in Shank Diameter

Fig. 5.15 Types of Cyclic Stresses

• S-N curve obtained from a rotating beam

test has completely reverse d stress state.

• Many stress histories will not have

completely reverse d stress state

5.4 Fluctuating stresses

Alternating stress

2

minmax

m

2

minmax

a

Mean stress

minmax rStress range

Stress ratio min

max

R

Amplitude ratio

m

aA

Fig. 5.16 Shear and Fatigue Failure of Wire (a) Shearing of Wire

(b) Bending of Wire (c) Unbending of Wire

5.5 Fatigue failure

• Early railroad cars moved on wheels rigidly attached

(shrunk) to a solid axle. The bearings were mounted

outside the wheels (Fig. a).

• The corresponding free­body diagram shows the

bearing supports of the beam shaft with vertical

forces acting at each wheel (Fig. b).

• At any instant, the axles is loaded in bending with

maximum stresses at top and bottom (Fig. c).

Fatigue in railroad axles (I)

• Because of rotation ̧the material at any point undergoes a

complete stress cycle every revolution (Fig. d).

• During operation, stress cycles accumulate rapidly, and

fracture may occur at either of the two bearings.

• Fatigue fracture surfaces often display two distinctly

different zones. The one section, often discolored by

corrosion, usually exhibits a pattern of lines or beach marks

(Fig. f).

Fatigue in railroad axles (II)

Fatigue in railroad axles (III)

• At times, the beach marks are so fine that they are visible

only magnification (such as is possible with an electron at

great microscope). Crack origin and direction of progression

are often indicated by these markings, which thus give a

clue to possible material flaws or inadequate design. The

other zone of the fracture usually has the bright, grainy

appearance of ductile rupture or fracture.

Mechanism of fatigue feature

• Crack initiation

• Crack propagation

• Fracture

Crack-initiation stage

• Some regions of geometric stress concentration in location

of time-varying that contains a tensile component.

• As the stresses at the notch oscillate, local yielding may

occur due to the stress concentration, even though the

nominal stress is below yield strength of the material.

• The localized plastic yielding causes distortion and creates

slip bands along the crystal boundaries of the material.

• As the stress cycles, additional slip bands occur and

coalesce into microscopic cracks.

• Because of their association with shear stress and

slip, microcracks are oriented with the maximum

shear stress. They may grows across several grains.

Crack propagation stage

• The sharp crack creates stress concentrations larger than

those of the original notch, and a plastic zone develops at the

crack tip each time a tensile stress opens the crack, blunting

its tip and reducing the effective stress concentration.

• This process continues as long as the local stress is cycling

from below the tensile yield to above the tensile yield at the

crack tip.

• The crack growth is due to tensile stress and the crack

grows along planes normal to the maximum tensile stress.

Fracture

• The growth of the cracks continues until a critical size is

reached such that one more application of the load brings

about instability and fracture.

Endurance limitChapter 5.6 endurance limit

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

maximum amplitude of completely reversed stress that the

standard specimen can sustain for an unlimited number of

cycles without fatigue failure.

cycles is considered as a sufficient number of cycles to

define the endurance limit.

610

Fig. 5.17 Specimen for Fatigue Test

Fig. 5.18 Rotating Beam

Subjected to Bending moment

(a) Beam,

(b) Stress Cycle at Point A

Fig. 5.19 Rotating Beam Fatigue Testing Machine

Fig. 5.20 S-N Curve for Steels

The S-N curve is the graphical representation of the stress

amplitude versus the number of the stress cycles (N) before

the fatigue failure on a log-log graph paper.

S-N curve

tS

tS

Fig. 5.21 Low and High Cycle Fatigue

5.7 Low cycle and high cycle fatigue

• Low cycle fatigue

Any fatigue failure, when the number of stress cycles are

less than 1000, is called low cycle fatigue.

This case is treated as the static condition and a larger

factor of safety is used and design on the basis of

ultimate strength or yield strength.

• High cycle fatigue

Any fatigue failure, when the number of stress cycles are

more than 1000, is called high cycle fatigue.

Components are designed on the basis of endurance limit

stress. S-N curve, Soderberg, Goodman are used in

design.

5.8 Notch sensitivity

specimen notched theoflimit endurance

specimen freenotch theoflimit endurancefK

)1( 1 tf KqK

Notch sensitivity is defined as the susceptibility of a material

to succumb to the damaging effects of stress raising notches in

fatigue loading.

The notch sensitivity factor is defined as

stress nominalover stress al theoreticof increae

stress nominalover stress actual of increaeq

Fatigue stress concentration factor is defined asfK

Fig. 5.22 Notch Sensitivity Charts (for Reversed Bending

and Reversed Axial Stresses)

Fig. 5.23 Notch Sensitivity Charts (for Reversed Torsional

Shear Stresses)

Chapter 5.9 Endurance limit ---Approximate estimation

The relationship between and is as follow:

where

= surface finish factor

= size factor

= reliability factor

= modifying factor to account for stress concentration

= endurance limit stress of a rotating beam specimen subjected to

reversed beam stress

= endurance limit stress of a particular mechanical component

subjected to reversed beam stress

'

ee SKKKKS dcba

bK

aK

cK

dK

eS

'

eS

'

eS

eS

There is an approximate relationship between the endurance

limit and the ultimate tensile strength of the material.

For steel,

For cast iron and cast steel,

For wrought aluminum alloys,

For cast aluminum alloys,

ute SS 5.0'

ut

'

e 4.0 SS

ut

'

e 4.0 SS

tSS u

'

e 3.0

Fig. 5.24 Surface Finish Factor

Surface Finish Factor

Size factor

The rotating beam specimen is small with 7.5 mm diameter.

The endurance limit reduces with increasing the size of the

component.

For bending and torsion, the value of size factor are given in

Table 5.2.

diameter d (mm)

d 7.5 1.0

7.5 < d 50 0.85

d > 50 0.75

bK

Table 5.2 Size factor

Fig. 5.25

Effective diameter is based on an equivalent circular cross-section.

Kuhuel assumes a volume of material that is stresses to 95 % of

maximum stress or above. As high stress volume.

Effective diameter

0766.0

95Ade

The effective diameter of any non-circular cross section is given by

222

95 0766.0]4

)95.0([ d

ddA

= portion of cross-sectional area of the non-cylindrical part that is

stresses between 95 % and 100 % of the maximum stress

= effective diameter of non-cylindrical part.ed

95A

The effective diameter is obtained by equating the volume of the material

stresses at and above 95 % of the maximum stresses to the equivalent

volume in the rotating beam specimens.

The area stressed above 95 % of the maximum stress is the area of a ring,

having an inside diameter of 0.95 d and outside 1.0 d .

Fig. 5.26 Area above 95% of Maximum Stress

For rectangular cross-section the effective

diameter bhde 808.0

dde 37.0

For non-rotating solid shaft, the effective

diameter

Reliability factor

Reliability R (%)

50 1.00

90 0.897

95 0.868

99 0.814

99.9 0.753

99.99 0.702

99.999 0.659

cK

The standard deviation of endurance limit test is 8 % of the

mean value.

The reliability factor is 1.0 for 50 %

reliability.

To ensure insure that more than 50 % of

the part will survive, the stress amplitude

on the component should be lower than

the tabulated value of endurance limit.

Reliability may be defined as the probability that a machine

part will perform its intended function without failure for its

prescribed design lifetime.

where p(x) is the probability density function, is the mean

value of the quantity, and the standard deviation.

- ]2

)([exp

2

1)(

)(1

1deviation standard

1mean

2

2

1

2

1

xx

xp

xn

xn

n

i

i

n

i

i

Modifying factor to account for stress concentration

f

dK

K1

The endurance limit is reduced due to stress concentration.

To apply the effect of stress concentration, the designer can

reduce the endurance limit by .dK

The endurance limit of a component subjected to the

fluctuating shear stresses is obtained from the endurance

limit in reversed bending ( ) using theories of failure.

From the maximum shear theory,

From the distortion energy theory,

ese 5.0 SS

ese 577.0 SS

seS

eS

5.10 Reversed stresses design for finite and infinite life

(1) Infinite life

Endurance limit is the criterion of failure. The amplitude

of stress should be lower than the endurance limit in order

to withstand the infinite number of cycles.

)( ,

)( fs

S

fs

S sea

ea

where are stress amplitude in the component and

are corrected endurance limit in reversed bending

and reversing torsion respectively.

aa ,

see SS ,

(2) Finite life

When the components is to be designed for finite life,

S-N curve (Fig. 5.27) can be used. It consists of a

straight line AB drawn from at cycles

to at cycles on a log-log paper.

310610

)9.0( utS

eS

Fig. 5.27 S-N Curve

At 1000 cycles:

Bending:

Axial bending:utm

utm

SS

SS

75.0

9.0

5.12 Soderberg and Goodman lines

• When stress amplitude is zero, the load is purely static

and criterion of failure is . These limits are plotted

on the abscissa.

• When the mean stress ( ) is zero, the stress is completely

reversing and the criterion of failure is endurance limit ,

that is plotted on the ordinate.

meS

a

utyt or SS

(1) Gerber line

A parabolic curve joining on the ordinate to on the abscissa.

(2) Soderberg line

A straight line joining on the ordinate to on the abscissa.

The equation of Soderberg line is given as:

(3) Goodman line

A straight line joining on the ordinate to on the abscissa

The equation of Goodman line is given as

eS

eS

ytS

utS

eS ytS

1e

a

yt

m SS

1e

a

ut

m SS

Fig. 5.39 Soderberg and Goodman Lines

Any combination of mean and alternating stress that lies on or

below Goodman line will have infinite life.

Goodman line is widely used in the criterion of fatigue failure

when the component is subjected to mean stress as well as

stress amplitude.

(1) Goodman line is safe from design consideration because it

is completely inside the failure points of test data.

(2) The equation of straight line is simple.

5.13 Modified Goodman Diagram

Goodman line is modified by combining fatigue failure with

fatigue by yielding.

The yield strength is plotted on both the axes-abscissa and

ordinate, and a yield line is constructed to join two points

to define failure by yielding.

The region OABC is called modified Goodman diagram. All

the points inside the modified Goodman diagram should cause

neither fatigue failure and yielding.

is the portion of Goodman line and is portion of

yield line.

DC

ytS

BACB

Modified Goodman Diagram

A line is drawn through on the ordinate and parallel to the

abscissa. The point of intersection of this line and yield line is

B. The area OABC represents the region of safety.

The region OABC is called modified Goodman diagram.

All the points inside the modified Goodman diagram should

neither fatigue failure nor yielding.

The point of intersection of lines is X. The point X

indicates the dividing line between the safe region and the

region of failure.

EOBA and

m

atan

The coordinates of point X ( ) represent the limiting

values of stress, that are used to calculate the dimensions of

component.

am SS ,

Fig. 5.40 Modified Goodman Diagram for Axial and Bending

Stresses

Fig. 5.41 Modified Goodman Diagram for Torsional Shear Stresses

The modified Goodman diagram for fluctuating torsional

shear stress is shown in Fig. 5.41.

Example 5.13

A transmission shaft of cold drawn steel 27Mn2 (

) is subjected to a fluctuating torque which

varies from -100 N-m to 400 N-m. The factor of safety is 2 and

expected reliability is 90 %. Neglecting the stress concentration,

determine the diameter of the shaft.

Solution:

,N/mm540 2

ut S2

yt N/mm 400 and S

897.0,reliablity %90for

85.0,mm 505.7 assuming

79.0

N/mm025)500(5.05.0

c

2

ut

'

K

Kd

K

kSS

b

a

e

04.59

67.1150

250

)(

)(tan

mN250]100400[2

1])()[(

2

1)(

mN150]100400[2

1])()[(

2

1)(

N/mm 1.173300577.0577.0

N/mm 88.8658.150577.0577.0

, theoryenerg distortion theusing

N/mm 58.150

250897.085.079.0

mt

at

mintmaxtat

mintmaxtmt

2

ytsy

2

ese

2

'

M

M

MMM

MMM

SS

SS

y

SKKKS ecbae

The modified Goodman diagram is shown in Fig. 5.44. The ordinate of

point X is .N/mm 88.86or 2

saS

Fig. 5.44

mm 83.30

44.43

1025016

N/mm 44.432

88.86

)(

.N/mm 88.86

3

3

2sa

2

sa

d

d

fs

S

S

a

Fig. 5.52 Gerber

line

5.14 Gerber line

1)( 2

ut

m

e

a S

S

S

SGerber equation is given as:

Gerber curve takes mean path through failure points. It is more

accurate than Goodman or Soderberg line.

5.15 Fatigue design under combined stresses

The general equation of distortion energy theory is as follows.

where are normal stresses in X, Y, Z directions and

are shear stresses in their respective planes.zyx , ,

zxyzxy , ,

The bending moment as well as torsional moment may have

two components – mean and alternating stresses. Such

problems involving combinational of stresses are solved by the

distortion energy theory of failure.

2

)](6)()()[( 222222

zxyzxyxzzyyx

In case of combined bending and torsional moments, there is a

normal stress accompanied by the torsional shear

stress .

0 zxyzzy

22 3 xyx

xyx

xaxm and

The mean and alternating component of are

respectively.

The mean and alternating component of are

respectively.

x

xy

xaxm and

22

m 3 xymxm

22

a 3 xyaxa

The Two stresses are used in the modified Goodman

diagram to design the component.

am ,

Example 5.20

A transmission shaft carries a pulley midway between the two

bearings. The bending moment at the pulley varies from 200

N-m to 600 N-m , as the torsional moment in the shaft varies

from 70 N-m to 200 N-m. The frequency of variation of

bending and torsioal moments are equal to the shaft speed. The

shaft is made of steel fee 400 (

). The corrected endurance limit of the shaft is

200 . Determine the diameter of the shaft using a

factor of safety of 2.

400 and ,N/mm540 yt

2

ut SS

2N/mm

2N/mm

2

3

3

33

abxa

2

3

3

33

mbxm

mintmaxtat

mintmaxtmt

minbmaxbab

minbmaxbmb

N/mm1018.2037

d

)1000200(32

d

)(32

N/mm1037.4074

d

)1000400(32

d

)(32

mN65]70200[2

1])()[(

2

1)(

mN135]70200[2

1])()[(

2

1)(

mN200]200600[2

1])()[(

2

1)(

mN400]200600[2

1])()[(

2

1)(

d

M

d

M

MMM

MMM

MMM

MMM

Solution

2

3

3

2

3

32

3

32

xya

2

xaa

2

3

3

2

3

32

3

32

xym

2

xmm

2

3

3

33

atxya

2

3

3

33

mtxym

N/mm)1033.2116

(

)1004.331

(3)1018.2037

(3

N/mm)1084.4244

(

)1055.687

(3)1037.4074

(3

N/mm1004.331

d

)100065(16

d

)(16

N/mm1055.687

d

)1000135(16

d

)(16

d

dd

d

dd

d

M

d

M

5.26

4986.084.4244

33.2116tan

m

a

The modified Goodman diagram is shown in Fig. 5.56. The

coordinate of point X are obtained by solving the following

two equations simultaneously.

Fig. 5.56

mm 29.33

38.571033.2116

N/mm 38.572

76.114

)(

N/mm 16.230

N/mm 76.114

4986.0tan

1540200

3

3

2a

2

m

2

a

m

a

ma

d

d

fs

S

S

S

S

S

SS

a

Exercise chapter 05

1. Prob. 5.10 page 201