stress concentrations - university of...

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 • 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

D

o

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 < ~0.005, say)

2a

o



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

31

32

−== o aaσσσ

0

32

2

31

2

4

4

2

2

22

==

++==

−==

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):

3

0

==

==

oyy

xxrr

σσσ

σσ

θθ

• The stress concentration factor for a circular hole inan infinite plate:

0

3

==

==

xyr

oyy

ττ

σσσ

θ

θθ

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

along the x-axis

3==o

yytK

σσ



• If a/D > 0.005 then the plate is “finite” and the Kirsch solution is no longer valid

• Stress concentration

D

o

• Stress concentration factors for a circular holes infinite plates have been measuredexperimentally for a range of a/D ratios (usually usingphotoelasticity), and tabulated inthe form of curve-fits in reference handbooks …required several yearsand many contributors

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 arein common use:

-based on the grossstress :maxσ

D

P

-based on the grossstress :

(σg remains constant as a increases)

-based on the netstress:

(σn increases as a increases)

Dt

PK g

g

yygt *

wheremax

== σσ

σ

)2(* re whe

max

aDt

PK n

n

yynt −

== σσ

σ

2a

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

aD

a

D

aK n

t


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 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:

)( )(2)(1 −= FFx ssxxσ

where:

0)(

)( )(2)(1

)(2)(1

=

+=

x

FFx

xy

ssyy

ssxx

τ

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 measure strain concentrations near stress risers:



• In general, even the smallest of commercial resistance strain gages are too large to measure strain concentrations

o

measure strain concentrations near stress risers:

…a poor experimental approach

o



• Instead, use a commercial “strip gage” and extrapolate experimental measurements to edge of stress riseredge of stress riser


Lab #5: Stress and Strain Concentrations


Corrections for Biaxial Rosettes With Differing Transverse Sensitivity Coefficients

yt

xt

myxt

ytomx

xto

xKK

KKK

−

−−−=

1

)1()1( ενενε

yxy KKK −−− )1()1( ενενyt

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 •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.


Lab #5: Stress and Strain Concentrations“Official Data” – Axial Strains

800

1000

1200

Axial Strains

Gage 1A

Gage 2A

-200

0

200

400

600

0 500 1000 1500 2000 2500 3000 3500 4000

Str

ain

(µε)

Applied Load (lbf)

Gage 2A

Gage 3A

Gage 4A

Gage 5A

Gage 6A

Gage 7A

Gage 8A

Gage 9A

Gage 10A


Lab #5: Stress and Strain Concentrations“Official Data” – Transverse Strains

20

40

60

Transverse Strains

Gage 1T

-80

-60

-40

-20

0

20

0 500 1000 1500 2000 2500 3000 3500 4000

Str

ain

(µε)

Applied Load (lbf)

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)

6

0

1

2

3

4

5

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



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.….fit using a 2nd-order polynomial:….fit using a 2 -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




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.….fit using an exponential:….fit 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




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..…fit using a power law.…fit 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



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.….fit using an polynomial and (1/normalized stress)….fit 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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