Machine Design Databook Episode 1 part 5 pot

STRESS INTENSITY FACTOR ORFRACTURE TOUGHNESS FACTOR The energy criterion approach in the fracture mechanics analysis: The energy release rate in case of a crack of length 2a in an infinit

FIGURE 4-22 Reproduced with permission Stress-concentration factor K
for notched flat bar in tension (R E Peterson,

‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol 23, Nos 2 to 7, 1951.)

(iii) Bar with shallow V-groove in tension for

(v) Bar with symmetrical U, semicircular

shallow grooves in bending (Fig 4-23)

24

35

266

377

FIGURE 4-23 Reproduced with permission

Stress-con-centration factor K
for notched flat bar in bending (R E.

Peterson, ‘‘Design Factors for Stress Concentration,’’

Machine Design, Vol 23, Nos 2 to 7, 1951.)

For stress-concentration factors for small grooves

in a shaft subjected to torsion

(o) Bar containing shoulders

(i) Bar with shoulders in tension (Fig 4-24)

TABLE 4-4

Stress-concentration factors for relatively small

grooves in a shaft subject to torsion,K

h 1 r Included angle of V, deg 0.5 1 3 5 2

24

35

0:85

ð4-28aÞor

K
¼ 1 þ

D

266

377

(ii) Bar with shoulders in bending (Fig 4-25)

(p) Press-fitted or shrink-fitted members (Table 4-5):

(i) Plain member

(ii) Grooved member

(iii) Plain member

(iv) Grooved member

(q) Bolts and nuts (Tables 4-6 and 4-7)

Bolt and nut of standard proportions

Bolt and nut having lip

FIGURE 4-25 Reproduced with permission Stress-concentration factor K
for stepped bar in bending (R E Peterson,

‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol 23, Nos 2 to 7, 1951.)

5:37D

d 4:8

h1r

24

35

0:85

ð4-29aÞor

K
¼ 1 þ

D

266

377

Index of sensitivity for repeated stress

Average index of sensitivity q

Heat-treated and drawn

(t) Eye bar:

For eye bar subjected to tensile load

Stress concentration factors for welds

(u) Notch sensitivity factors (Table 4-9):

(i) Notch sensitivity factor for normal stress

For index of sensitivity for repeated stresses

(ii) Fatigue stress concentration factor for

normal stress

(iii) Notch sensitivity factor for shear stress

(iv) Fatigue stress-concentration factor for shear

stress

STRESS CONCENTRATION IN FLANGED PIPE

SUBJECTED TO AXIAL EXTERNAL FORCE

The stress in the pipe due to external load F (Fig

where
f ¼ depends on the distance x from the

flange of the pipe, MPa (psi)

fm¼ maximum stress at x ¼ 0, MPa (psi)

A ¼ area of the cross section of pipe, m2(in2)

The value of constant

For plot of the stress ratio
f

fmversusx

FIGURE 4-25B Stress concentration region in flanged pipe under axial external force F.

Courtesy: Douglas C Greenwood, Engineering Data for Product Design, McGraw-Hill Publishing Company, New York, 1961.

h ¼ thickness of pipe, m (in)

 ¼ Poisson’s ratio of materialRefer to Fig 4-25B

REDUCTION OR MITIGATION OF STRESS CONCENTRATIONS

In designing a machine part, one has to take into consideration the stress concentration occurring in such parts andeliminate or reduce stress concentration Fig 4-25C shows various methods used to reduce stress concentration.Stream line flowing analogy in a channel can be applied to force flow lines of a flat plate without any type of flowsubject to uniform uniaxial tensile stress
as shown in Fig 4-25C i(a) The stream line flow of water or any fluid is

smooth and straight as shown in Fig 4-25C i(a) If there is any obstruction such as a heavy iron ball or a pipe orstone boulder in the path of flow of water, the flow of water or fluid will not be smooth and straight as shown inFig 4-25C i(e) Similarly the force flow lines will not be straight as in case of plate with a circular or elliptical or anyshape of holes in a plate as shown in Figs 4-25C i(b), i(c), i(d) and i( f ) By providing some geometric changes,abrupt change of force-flow lines are smoothened Fatigue strength of parts with stress raiser can be increased

by cold working operation such as shot peening or pressing by balls which creates a nature of stress in thinlayers of the part just opposite to the one induced in it Press fit stress concentration can be reduced by makingthe gripping portion conical in case of hardening steel parts Nitriding and plating the parts eliminate the corrosioneffect, which combined with stress concentration reduces the fatigue strength of the machine part

(i) Plates:

(ii) Stepped shafts subject to tensile force:

(iii) Shafts with narrow collar, cylindrical holes and grooves subject to tensile force:

(iv) Shafts subject to bending and torque:

(v) Screws and nuts under torque:

(vi) Keyways in shafts subject to torque:

(viii) Flate plate with and without asymmetrically reinforced circular cutout subjected to uniform uniaxial stress:

FIGURE 4-25C Mitigation of stress concentration in machine members.

s

h

(a)

Forces acting on a Gear Tooth Profile

due to Normal Force F n

fillet with less stress concentration

Compression side of root fillet gear tooth with more fringes compare to tension side root fillet stress analysis

of gear teeth in contact under load showing photoelastic fringes by using the results of photoelastic experiment

ρt

β

α

α

Fringe pattern of gear teeth

in contact showing stress concentration at root and point

of contract

(b)

STRESS INTENSITY FACTOR OR

FRACTURE TOUGHNESS FACTOR

The energy criterion approach in the fracture

mechanics analysis:

The energy release rate in case of a crack of length 2a

in an infinite plate subject to tensile stress at infinity

(Fig 4-26A)

The energy release rate is defined as the rate of change

in potential energy with crack area for a linear elastic

material

The critical energy release rate

FIGURE 4-26A A flat infinite plate with a through thickness crack subject to tensile stress at infinity.

The stress intensity factor for a centrally located

straight crack in an infinite plate subjected to uniform

uniaxial tensile stress
perpendicular to the plane of

where G ¼ energy release rate

E ¼ modulus of elasticity, GPa (psi)

a ¼ half crack length, mm (in)

Gc¼

2

fac

where
f ¼ failure stress, MPa (psi)

Gc¼ material resistance to fracture orcritical fracture toughness

KI¼pffiffiffi
pffiffiffia

ð4-34cÞ

KIc is the critical stress intensity factor for staticloading and plane-strain conditions of maximum con-straints and is also referred to as the fracture tough-ness factor of the material at the onset of rapidfracture and has dimension of (stress ffiffiffiffiffiffiffiffiffiffiffiffiffi

length

p), i.e.MPa ffiffiffiffi

m

p(kpsi ffiffiffiffiin

p)

G ¼KI

Three modes of loading to analyse stress

fields in cracks:

FIGURE 4-26B Three modes of loading for deformation of crack tip.

First mode of loading and stress components

at crack tip, Fig 4-26B (a):

The localized stress components at the vicinity of the

‘‘opening mode’’ or ‘‘mode I’’ crack tip in a flat plate

subjected to uniform applied stress
at infinity from

the theory of fracture mechanics (Fig 4-26C)

The crack tip displacement fields for ‘‘first mode’’

(Mode I) in case of linear elastic, isotropic materials

FIGURE 4-26C State of stress in the vicinity of a crack tip.



ð4-35bÞ

z¼ ð
xþ
yÞ for plane strain ð4-35cÞ

xy¼ KffiffiffiffiffiffiffiI2r

ffiffiffiffiffiffir2

rcos2

ffiffiffiffiffiffir

2

rsin2

 ¼ 3  4 for plane strain

 ¼ ð3  Þ=ð1 þ Þ for plane stress

Second mode of loading and stress

components in the vicinity of crack tip,

Fig 4-26B (b):

The localized stress components at the vicinity of the

‘‘second mode’’ or ‘‘sliding mode’’ crack tip in a flat

plate subjected to in-plane shear, Fig 4-26B (b)

The crack tip displacement fields for the ‘‘second

mode’’ (Mode II) in case of linear elastic, isotropic

materials

Third mode of loading and stress components

in the vicinity of crack tip, Fig 4-26B (c):

The localized stress components at the vicinity of the

‘‘third mode’’ or ‘‘tearing mode III’’ crack tip in a flat

plate subjected to out-of-plane shear, Fig 4-26B (c),

in case of linear elastic, isotropic materials

The crack tip displacement field for the ‘‘third mode’’

(Mode III) in case of linear elastic, isotropic materials

Stress intensity factor:

The stress intensity factor for a center cracked tension

plate (CCT), according to Fedderson (Fig 4-27a)

The stress intensity factor for a double edge cracked

plate according to Keer and Freedman (Fig 4-27b)

cos



32



ð4-35jÞ

z¼ ð
x
yÞ for plane strain ð4-35lÞ

ux¼KII2G

ffiffiffiffiffiffir

2

rsin2

ffiffiffiffiffiffir

2

rcos2



  1  2 sin2

2

ð4-35oÞ

xz¼ KffiffiffiffiffiffiffiIII2r

p sin

yz¼ KffiffiffiffiffiffiffiIII2r

p cos

uz¼KIIIG

ffiffiffiffiffiffir

2

rsin

w ¼  ¼ 0

KI¼
p ffiffiffiffiffiffia sec

a2b



ð4-35sÞ

Ki¼
p ffiffiffiffiffiffia 1:12  0:61

ab



þ 0:13

ab

3





1ab

1=2

ð4-35tÞ

The stress intensity factor for the plate with a single

edge crack, according to Gross, Srawley and Brown

(Fig 4-27c)

The stress intensity factor for single edged cracked

plate/specimen subjected to bending (Mb) (Fig

4-27d)

Stress intensity factor for the case of angled

crack (Fig 4-27A):

FIGURE 4-27A Through crack in an infinite plate for the

general case where the crack plane is inclined at 90 8  

angle from the applied normal stress
acting at infinity.

The stress intensity factors for Modes I and II

KI¼
p ffiffiffiffiffiffia 1:12  0:23

ab



þ 10:6

ab

2

 21:7

ab

3

þ 30:4

ab

4

ð4-35uÞ

KI¼
p ffiffiffiffiffiffia 1:112  1:40

ab



þ 7:33

ab

2

 13:08

ab

3

þ 14:0

ab

4

ð4-35vÞ

where KIð0Þis the Mode I stress intensity factor when

 ¼ 0

Equations for stress and displacement

components in terms of polar coordinates:

The localized stress components at the vicinity of

Mode I crack tips in terms of polar coordinates

The crack tip displacement fields for ‘‘first mode’’

(Mode I) in case of linear elastic, isotropic materials

The localized stress components at the vicinity of

Mode II crack tip in terms of polar coordinates

The crack tip displacement fields for Mode II

r¼ KI

4 ffiffiffiffiffiffiffi

2rp



5 cos

2 cos32



3 cos

2þ cos32

sin

2þ sin32



ð4-36eÞ

where1¼ 0 for plane stress and 1is Poisson’s ratio,

, for plane strain These singular fields only apply as

r ! 0

ur¼KI2E

ffiffiffiffiffiffir2

rð1 þ Þ

ð2  1Þ cos

2 cos32

ð4-36gÞ

u¼KI2E

ffiffiffiffiffiffir2

rð1 þ Þ



ð2  1Þ sin

2þ sin32

ð4-36hÞ

uz¼ 



2zE

 ¼ ð3  4Þ, 1¼ , and 2¼ 0 for plain strain

KI is given by Eq (4-36a)

ffiffiffiffiffiffir

2

rð1 þ Þ

u¼KII2E

ffiffiffiffiffiffir

2

rð1 þ Þ

The localized stress components and crack tip

dis-placement fields for Mode III in terms of polar

coordinates

The critical applied tensile stress necessary for crack

extension according to Griffith theory for brittle

metals

The modified Griffith’s equation for a small amount

of plastic deformation according to Orowan which

can be applied to ductile materials at low

tem-perature, high strain rate and localized geometric

constraint

The elastic energy release rate for Mode I

The elastic energy release rate for Mode II

The elastic energy release rate for Mode III

The stress-intensity factor for a centrally located

straight crack in an infinite plate subjected to uniform

shear stress

The stress-intensity magnification factor for a

cen-trally located straight crack of length 2a in a flat

plate whose length 2h and width 2b are very large

compared with the crack length subjected to uniform

uniaxial tensile stress

For stress-intensity magnification factors of plates

with straight crack located at various positions in

the plate and cylinders subjected to various types of

rate of loadings and for various values of a=b, a=d,

a=h, a=ðro riÞ, and other ratios

The factor of safety

ffiffiffiffiffiffir

2

rsin

c/

ffiffiffiffiffiffiffiffiEUa

r

ð4-36rÞwhere
c¼ critical applied stress

r

ð4-36sÞwhere p ¼ plastic deformation energy per unit areafor metallic solid, p  U

GI¼



1 2E

FIGURE 4-28 Stress intensity magnification factor

K I =pffiffiffi
pffiffiffia

for various ratios a=b of a flat plate with a

cen-trally located straight crack under the action of uniform

uni-axial tensile stress
.

FIGURE 4-30 Stress intensity magnification factor

K I =pffiffiffi
pffiffiffiafor an edge straight crack in a flat plate subjected

to uniform uniaxial tensile stress
for solid curves there are

no constraints to bending; the dashed curve was obtained

with bending constraints added.

FIGURE 4-29 Stress intensity magnification factor

K I =pffiffiffi
pffiffiffia

for an off-center straight crack in a flat plate jected to uniform unidirectional tensile stress
; solid curves are for the crack tip at A; dashed curves for tip at B.

sub-FIGURE 4-31 Stress intensity magnification factor

K I =pffiffiffi
pffiffiffiafor a rectangular cross-sectional beam subjected

to bending M b

FIGURE 4-32 Stress intensity magnification factor

K I =pffiffiffi
pffiffiffia

for a flat plate with a centrally located circular

hole with two straight cracks under uniform uniaxial tensile

FIGURE 4-34 Stress intensity magnification factor K I =pffiffiffi
pffiffiffia

for a cylinder subjected to internal pressure p i having a radial crack in the longitudinal direction of depth a Use equation of tangential stress of thick cylinder subjected to internal pressure to calculate the stress
 at r ¼ r o

Critical crack length

For values of critical stress-intensity factor (KIc) for

some engineering materials

REFERENCES

1 Lingaiah, K., Solution of an Asymmetrically Reinforced Circular Cut-out in a Flat Plate Subjected toUniform Unidirectional Stress, Ph.D Thesis, Department of Mechanical Engineering, University ofSaskatchewan, Saskatoon, Sask., Canada, 1965

2 Lingaiah, K., W P T North, and J B Mantle, ‘‘Photoelastic Analysis of an Asymmetrically ReinforcedCircular Cut-out in a Flat Plate Subjected to Uniform Unidirectional Stress,’’ Proc SESA, Vol 23, No 2(1966), p 617

3 Peterson, R E., ‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol 23, No 27, Pentagon lishing, Cleveland, Ohio, 1951

Pub-4 Lingaiah, K., ‘‘Effect of Contact Stress on Fatigue Strength of Gears,’’ M.Tech Thesis, Indian Institute ofTechnology, Kharagpur, India, 1958

5 Lingaiah, K., ‘‘Photoelastic Stress Analysis of Gear Teeth Under Load,’’ Department of MechanicalEngineering, University Visveswaraya College of Engineering, Bangalore University, Bangalore, 1980

6 Lingaiah, K., and B R Narayana Iyengar, Machine Design Data Handbook, Vol I (SI and Customary MetricUnits), Suma Publishers, Bangalore, India, 1986

7 Lingaiah, K., Machine Design Data Handbook, Vol II (SI and Customary Metric Units), Suma Publishers,Bangalore, India, 1986

Plane-strain fracture toughness or stress intensity factorKIcfor some engineering materials

Material K Ic Yield strength, xy Critical crack length,a c Previous designation UNS designation MPa ffiffiffiffi

m p kpsi ffiffiffiffi in

8 Lingaiah, K., Machine Design Data Handbook (SI and Customary US Units), McGraw-Hill Publishing pany, New York, 1994.

Com-9 Aidad, T., and Y Terauchi, ‘‘On the Bending Stress in a Spur Gear,’’ 3 Reports, Bull JSME, Vol 5 (1962),

14 Carlson, R L., and G A Kardomateas, An Introduction to Fatigue in Metals and Composites

15 Anderson, T L., Fracture Mechanics—Fundamentals and Application, 2nd edition, CRC Press, New York,1995

16 Fedderson, C., ‘‘Discussion’’, in Plane Strain Crank Toughness Testing of High Strength Metallic Materials,ASTM STP410, American Society for Testing Materials, Philadelphia (1967), p 77

17 Keer, L M., and J M Freedman, ‘‘Tensile Strip with Edge Cracks,’’ Int J Engineering Science, Vol 11(1973), pp 1965–1075

18 Gross, B., and J E Srawley, ‘‘Stress Intensity Factors for Bend and Compact Specimens,’’ EngineeringFracture Mechanics, Vol 4 (1972), pp 587–589

19 Gross, B., J E Srawley, and W E Brown Jr., Stress Intensity Factors for a Single Edge Notch TensionSpecimen by a Boundary Collocation of a Stress Function, NASA Technical Note D-2395, 1964

20 Damage Tolerant Design Handbook, MICIC-HB-01, Air Force Materials Laboratory, Wright-Patterson AirForce Base, Ohio, December 1972, and supplements