Stress Concentrations - University of Washingtoncourses.washington.edu/mengr556/StressConcentrations.pdfStress Concentrations • A “stress concentration” refers to an area in

Prof. M. E. Tuttle University of Washington

Stress Concentrations

• A “stress concentration” refers to an area in a

object where stress increases over a very short

distance (i.e., where a high stress gradient exists)

• Stress concentrations typically occur due to some

localized change in geometry (near holes, filets,

corners, grooves, cracks, etc)

• These changes in geometry are often called “stress

risers”


Stress Concentration Near a Circular Hole

• In 1898 Ernst Kirsch (a German

engineer) published a solution for

the elastic stresses near a circular

hole in an isotropic “infinitely

large” thin plate (the Kirsch

solution is derived in Sec 3.13 of

the Shukla and Dally textbook)

• In practice, a thin plate can be

considered to be “infinitely large”

if the hole diameter is small

compared to the in-plane plate

dimensions (if a/D < ~0.05, say)

D

2a

o

o



• Stresses along the x-axis in an infinite plate predicted by the

Kirsch solution:

0

32

2

31

2

4

4

2

2

2

3

2

2

==

++==

−==

xyr

oyy

oxxrr

x

a

x

a

x

a

x

a

ττ

σσσ

σσσ

θ

θθ

Figure 3.6: Distribution of σxx/σo and σyy/σo

along the x-axis



• Stresses at the edge of the hole (at x = a):

• The stress concentration

factor for a circular hole in

an infinite plate:

0

3

0

==

==

==

xyr

oyy

xxrr

ττ

σσσ

σσ

θ

θθ

Figure 3.6: Distribution of σxx/σo and σyy/σo

along the x-axis

3==o

yytK

σσ



• If a/D > 0.05 then the plate

is “finite” and the Kirsch

solution is no longer valid

• Stress concentration

factors for a circular holes in

finite plates have been measured

experimentally for a range of

a/D ratios (usually using

photoelasticity), and tabulated in

the form of curve-fits in reference

handbooks …required several years

and many contributors

D

o

o

2a



• Example: Wahl, A.M., and Beeuwkes, R.,

“Stress Concentration Produced by Holes and

Notches”, Transaction of the ASME; Applied

Mechanics, Vol 56 (11) , 1930



• Two different definitions of the

stress concentration factors are

in common use:-based on the gross stress :

(σg remains constant as a increases)

-based on the net stress:

(σn increases as a increases)

Dt

PK g

g

yygt

* where

max

== σσ

σ

)2(* re whe

max

aDt

PK n

n

yynt −

== σσ

σ

2a

D

P

P



• Example: from

Roark’s Formulas for Stress

and Strain (2002): D

o

o

2a

3

2

253.1

267.3

214.300.3

−

+

−=

D

a

D

a

D

aK n

t


Tabulated Stress Concentration Factors

• Stress concentration factors for many types of “stress risers”

have been tabulated…for example:

-Young, W.C., and Budynas, R.G., Roark’s Formulas for Stress and

Strain, 7th edition, McGraw-Hill, (2002)

- online tabulations: http://www.amesweb.info/


Stress Concentration Near an Elliptical Hole

• In 1913 Charles Inglis (a British

mathematician) published a

solution for the elastic stresses near

an elliptical hole in an isotropic

infinitely large thin plate (the Inglis

solution is discussed in Sec 4.2)

• In this case the stress concentration

depends on both the aspect ratio of

the hole (a/b) and on the size of the

plate



• For an infinite plate the stresses

along the x-axis

(i.e., for x ≥ a, y = 0) are given by:

where:

0)(

)(

)(

)(2)(1

)(2)(1

=

+=

−=

x

FFx

FFx

xy

ssyy

ssxx

τ

σ

σ

ba

bambaBm

B

x

B

xs

ms

ms

ms

m

ms

mF

ms

mF ss

+−=+=−

+=

−−

−−+

−−+=

−++=

)(2

1

22

311

11

2

)1(21

2

2

2

2

22

2

)(22)(1σσ



• For a finite plate:

babaC

babaC

babaC

babaC

D

aC

D

aC

D

aCCK n

t

/011.4/204.5270.2

/494.4/183.5621.3

/483.2/021.0351.0

/000.2/000.0000.1

222

4

3

2

1

3

4

2

321

−+−=

+−=

−−−=

++=

+

+

+=


Measuring Stress Concentrations Using Strain

Gages

• In general, even the smallest

of commercial resistance

strain gages are too large to

measure strain concentrations

near stress risers:



Gages



Gages

• In general, even the smallest

of commercial resistance

strain gages are too large to

measure strain concentrations

near stress risers:

…a poor experimental

approach

o

o



Gages

• Instead, use a commercial

“strip gage” and extrapolate

experimental measurements to

edge of stress riser


Pseudo Lab #6: Stress and Strain Concentrations



0.065 in

0.080 in

(typical)

1 2 3 4 5 6 7 8 9 10

Axial gages

M-M gage type SA-13-031PJ-120

Gage factor = 2.12, Kt = 2.0%

0.065 in

0.080 in

(typical)

1 2 3 4 5 6 7 8 9 10

Transverse gages

M-M gage type EA-13-031MF-120

Gage factor = 2.09, Kt = 1.2%


Corrections for Biaxial Rosettes With Differing

Transverse Sensitivity Coefficients

yt

xt

myxt

ytomx

xto

xKK

KKK

−

−−−=

1

)1()1( ενενε

yt

xt

mxyt

xtomy

yto

yKK

KKK

−

−−−=

1

)1()1( ενενε

=mymx εε , strains measured in the x- and y- directions

=yt

xt KK , Transverse sensitivity coefficients for gages

in the x- and y- directions

MM Tech-Note 509 “Errors Due to Transverse Sensitivity in Strain Gages”



Goals:

•To compare stress distributions measured near an

elliptical hole in a finite thin plate to those predicted

for an infinite thin plate, and

•To compare the stress concentration factor measured

for an elliptical hole in a finite thin plate to the value

expected from a reference handbook.


Pseudo Lab #6: Stress and Strain Concentrations“Official Data” – Axial Strains

-200

0

200

400

600

800

1000

1200

0 500 1000 1500 2000 2500 3000 3500 4000

Str

ain

(µε

)

Applied Load (lbf)

Axial Strains

Gage 1A

Gage 2A

Gage 3A

Gage 4A

Gage 5A

Gage 6A

Gage 7A

Gage 8A

Gage 9A

Gage 10A


Pseudo Lab #6: Stress and Strain Concentrations“Official Data” – Transverse Strains

-80

-60

-40

-20

0

20

40

60

0 500 1000 1500 2000 2500 3000 3500 4000

Str

ain

(µε

)

Applied Load (lbf)

Transverse Strains

Gage 1T

Gage 2T

Gage 3T

Gage 4T

Gage 5T

Gage 6T

Gage 7T

Gage 8T

Gage 9T

Gage 10T



Goal 1: Compare stress distributions measured near an elliptical

hole in a finite thin plate to those predicted for an infinite thin plate

(Suggestion: compare normalized stresses)

0

1

2

3

4

5

6

0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45

σ yy/σ

net

Position along x-axis (in)

Inglis Solution Measured Note: as

mentioned

during class

lecture, the

measured

values shown

here are

fictitious



Goal 2: To compare the stress concentration factor measured for an

elliptical hole in a finite thin plate to the value expected from a

reference handbook.

(Suggestion: extrapolate a curve fit)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5 1 1.5 2

Normalized axial stress





reference handbook.

….fit of fictitious data using a 2nd-order polynomial:

y = 4.84x2 - 13.269x + 10.148

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5 1 1.5 2






reference handbook.

….fit of fictitious data using an exponential:

y = 5.2287e-1.103x

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5 1 1.5 2






reference handbook.

.…fit of fictitious data using a power law

y = 1.7238x-1.317

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5 1 1.5 2

Normalized Axial Stress





reference handbook.

….fit of fictitious data using an polynomial and (1/normalized

stress)

y = 1.2864x3 - 5.6909x2 + 8.6649x - 3.6261

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2

1/Normalized Axial Stress

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Stress Concentrations - University of Washingtoncourses.washington.edu/mengr556/StressConcentrations.pdfStress Concentrations • A “stress concentration” refers to an area in