stress concentrations - university of...
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![Page 1: Stress Concentrations - University of Washingtoncourses.washington.edu/mengr556/StressConcentrations.pdfStress Concentrations • A “stress concentration” refers to an area in](https://reader033.vdocuments.us/reader033/viewer/2022060210/5f0487e37e708231d40e6e1b/html5/thumbnails/1.jpg)
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”
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Prof. M. E. Tuttle University of Washington
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
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Prof. M. E. Tuttle University of Washington
Stress Concentration Near a Circular Hole
• 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
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Prof. M. E. Tuttle University of Washington
Stress Concentration Near a Circular Hole
• 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
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Prof. M. E. Tuttle University of Washington
Stress Concentration Near a Circular Hole
• 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solution is no longer valid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
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Prof. M. E. Tuttle University of Washington
Stress Concentration Near a Circular Hole
• Example: Wahl, A.M., and Beeuwkes, R.,
“Stress Concentration Produced by Holes and
Notches”, Transaction of the ASME; Applied
Mechanics, Vol 56 (11) , 1930
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Prof. M. E. Tuttle University of Washington
Stress Concentration Near a Circular Hole
• 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
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Prof. M. E. Tuttle University of Washington
Stress Concentration Near a Circular Hole
• 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
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Prof. M. E. Tuttle University of Washington
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/
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Prof. M. E. Tuttle University of Washington
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
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Prof. M. E. Tuttle University of Washington
Stress Concentration Near an Elliptical Hole
• 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
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Prof. M. E. Tuttle University of Washington
Stress Concentration Near an Elliptical Hole
• 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
−+−=
+−=
−−−=
++=
+
+
+=
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Prof. M. E. Tuttle University of Washington
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:
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Prof. M. E. Tuttle University of Washington
Measuring Stress Concentrations Using Strain
Gages
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Prof. M. E. Tuttle University of Washington
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:
…a poor experimental
approach
o
o
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Prof. M. E. Tuttle University of Washington
Measuring Stress Concentrations Using Strain
Gages
• Instead, use a commercial
“strip gage” and extrapolate
experimental measurements to
edge of stress riser
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Prof. M. E. Tuttle University of Washington
Pseudo Lab #6: Stress and Strain Concentrations
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Prof. M. E. Tuttle University of Washington
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%
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Prof. M. E. Tuttle University of Washington
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”
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Prof. M. E. Tuttle University of Washington
Pseudo Lab #6: Stress and Strain Concentrations
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.
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Prof. M. E. Tuttle University of Washington
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
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Prof. M. E. Tuttle University of Washington
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
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Prof. M. E. Tuttle University of Washington
Pseudo Lab #6: Stress and Strain Concentrations
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
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Prof. M. E. Tuttle University of Washington
Pseudo Lab #6: Stress and Strain Concentrations
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
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Prof. M. E. Tuttle University of Washington
Pseudo Lab #6: Stress and Strain Concentrations
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 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
Normalized axial stress
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Prof. M. E. Tuttle University of Washington
Pseudo Lab #6: Stress and Strain Concentrations
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 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
Normalized axial stress
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Prof. M. E. Tuttle University of Washington
Pseudo Lab #6: Stress and Strain Concentrations
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 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
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Prof. M. E. Tuttle University of Washington
Pseudo Lab #6: Stress and Strain Concentrations
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 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