stress concentrations - university of...
Post on 10-Jun-2020
3 Views
Preview:
TRANSCRIPT
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”
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
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
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
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solution is no longer valid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
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
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
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
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/
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
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
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
−+−=
+−=
−−−=
++=
+
+
+=
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:
Prof. M. E. Tuttle University of Washington
Measuring Stress Concentrations Using Strain
Gages
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
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
Prof. M. E. Tuttle University of Washington
Pseudo Lab #6: Stress and Strain Concentrations
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%
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”
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.
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
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
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
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
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
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
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
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
top related