measurement of residual stress/strain
TRANSCRIPT
Measurement of Residual Stress/Strain (Using Strain Gages and the Hole Drilling Method)
Summary of Discussion in Section 8.9
The Hole Drilling Method Is Based On:
(a) Stress transformation equations
( ) ( )θθτθθσστ
θθτθσθσσ
θθτθσθσσ
22''
22''
22''
sincossincos
sincos2cossin
sincos2sincos
−+−=
−+=
++=
xyxxyyyx
xyyyxxyy
xyyyxxxx
The Hole Drilling Method Is Based On:
(a) Stress transformation equations…label the new axes “r” and “θ ”(instead of x' and y')
( ) ( )θθτθθσστ
θθτθσθσσ
θθτθσθσσ
θ
θ22
22
22
sincossincos
sincos2cossin
sincos2sincos
−+−=
−+=
++=
xyxxyyr
xyyyxx
xyyyxxr
The Hole Drilling Method Is Based On:
(b) Stresses in a Uniformly-Loaded Thin Plate
The Hole Drilling Method Is Based On:
(b) Stresses in a Uniformly-Loaded Thin PlateThe state of stress is identical at all points within the
plate (ignoring possible stress concentrations near the
loaded ends)
o
o
+x
+y
The Hole Drilling Method Is Based On:
(b) Stresses in a Uniformly-Loaded Thin PlateDefine an x-y coordinate system at the center of the
plate:
σxx = 0
σyy = σo
τxy = 0
The Hole Drilling Method Is Based On:
(b) Stresses in a Uniformly-Loaded Thin PlateRotate the stress element along the
circumference of an (imaginary) circle of
radius r
o
o
+x
+y
r
The Hole Drilling Method Is Based On:
(b) Stresses in a Uniformly-Loaded Thin PlateRotate the stress element along the
circumference of an (imaginary) circle of
radius r…at any angle θ :
Since σxx = τxy = 0, σyy = σo
( ) ( )θθτθθσστ
θθτθσθσσ
θθτθσθσσ
θ
θ22
22
22
sincossincos
sincos2cossin
sincos2sincos
−+−=
−+=
++=
xyxxyyr
xyyyxx
xyyyxxr
θθστθσσ
θσσ
θ
θsincos
cos
sin
2
2
or
o
or
==
=
o
o
+x
+y
r
The Hole Drilling Method Is Based On:
(b) Stresses in a Uniformly-Loaded Thin PlateUsing trig identities:
The stress components can be written:
eqs (e), pg 69
eqs (8.44), pg 244
( )
( )
θθθ
θθ
θθ
2sin2
1sincos
2cos12
1cos
2cos12
1sin
2
2
=
+=
−=
θστ
θσσ
θσσ
θ
θ
2sin2
1
)2cos1(2
1
)2cos1(2
1
or
o
or
=
+=
−=
o
o
+x
+y
r2a
The Hole Drilling Method Is Based On:
(c) The Kirsch Solution
• (Derived in section 3.13): The stresses induced at
any point (r,θ ) in an infinitely large thin plate with
hole of radius a, subjected to a remote uniaxial
stress σo, are given by (where r ≥ a):
eqs (3.42)
−
+=
++
+=
−+
−=
θστ
θσσ
θσσ
θ
θ
2sin13
12
2cos3
112
2cos13
112
2
2
2
2
4
4
2
2
2
2
2
2
r
a
r
a
r
a
r
a
r
a
r
a
or
o
or
(…an Aside..) 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
(…an Aside…) 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
The Hole Drilling Method
Assume: A uniaxial residual stress exists in a
thin plate (Note a uniaxial residual stress field
is unusual…will generalize this discussion
later…)
σo
σo
+x
+y
r
εa
εb
εc
The Hole Drilling Method
Assume: A uniaxial residual stress exists in a
thin plate (Note a uniaxial residual stress field
is unusual…will generalize this discussion
later…)
Hole-drilling Procedure:
a) Special 3-element strain gage rosette
bonded to the plate…Note:
- gage elements arranged at constant
radius r in a circular pattern
- gage circuits are balanced in this
condition…since residual stresses
are present, the measured zero strain
does not correspond to zero stress
The Hole Drilling Method
Assume: A uniaxial residual stress exists in a
thin plate (Note a uniaxial residual stress field
is unusual…will generalize this discussion
later…)
Hole-drilling Procedure:
a) Special 3-element strain gage rosette
bonded to the plate…Note:
- gage elements arranged at constant
radius r in a circular pattern
- gage circuits are balanced in this
condition…since residual stresses
are present, the measured zero strain
does not correspond to zero stress
- A hole with radius a is drilled through the
center of the rosette, releasing the residual
stresses that existed in the hole volume
The Hole Drilling Method
• The strain gages respond to the change in the stress field, which is given
by:
(eqs 3.42) – (eq 8.44)
• Making the substitution and performing algebraic simplifications, the
stress field that the strain gages respond to is given by:
eqs (8.45)
−−=
+=
−+−=
θστ
θσσ
θσσ
θ
θ
2sin23
2
2cos3
12
2cos43
12
2
2
2
2
2
2
2
2
2
2
2
2
r
a
r
a
r
a
r
a
r
a
r
a
or
o
or
uniform stress plate with hole
The Hole Drilling Method
• The change in radial and tangential strains caused by (careful!) drilling of
the hole can be obtained by substituting the change in stress (i.e., eqs 8.45)
into Hooke’s Law for plane stress:
• Making this substitution and performing algebraic simplifications:
eqs (8.46)
( ) ( ) 1
1rrr
EEνσσενσσε θθθ −=−=
+−++=
+−++−=
νθνθνσε
νθθνσε
θ1
2cos42cos
31
2
)1(
1
2cos42cos
31
2
)1(
2
2
2
2
2
2
2
2
r
a
Er
a
r
a
Er
a
o
or
The Hole Drilling Method
• The change in radial and tangential strains caused by (careful!) drilling of
the hole can be obtained by substituting the change in stress (i.e., eqs 8.45)
into Hooke’s Law for plane stress:
• Making this substitution and performing algebraic simplifications:
eqs (8.46)
( ) ( ) 1
1rrr
EEνσσενσσε θθθ −=−=
+−++=
+−++−=
νθνθνσε
νθθνσε
θ1
2cos42cos
31
2
)1(
1
2cos42cos
31
2
)1(
2
2
2
2
2
2
2
2
r
a
Er
a
r
a
Er
a
o
or
Radial strains measured by
the 3-element rosette, if
residual stress is uniaxial
and aligned with y-axis
The Hole Drilling MethodPractical Application
• The residual stress field is usually biaxial, involving:
• two unknown principal residual stresses
• unknown orientation of principal axis,
RRσ21 and σ
)(or 21 θθ
Figure 8.13
The Hole Drilling MethodPractical Application
• The predicted strains induced by a uniaxial stress σo (eqs
8.26) can be used to separately calculate:
• strains induced by principal stress
• strains induced by principal stress (acting at 90º w/r/t )
R1σR2σ R
1σ
The Hole Drilling MethodPractical Application
• The predicted strains induced by a uniaxial stress σo (eqs
8.26) can be used to separately calculate:
• strains induced by principal stress
• strains induced by principal stress (acting at 90º w/r/t )
• Applying the principal of superposition, using Hooke’s law
and the strain transformation equations we find:
R1σR2σ R
1σ
( ) ( )
( ) ( )
++
−
+−=
+−=
−+−=
−+−−+=
−+−++=
ννν
εεεεεθ
εεεεεεσ
εεεεεεσ
1
43
2
1
2
1
22tan
4
2
4
4
2
4
22
2
2
1
22
212
22
211
r
a
r
a
EC
r
a
EC
CC
CC
AC
CBA
cBBACAR
cBBACAR
eqs (8.53)
Fig 8.14: Residual
Stress (Strain) Gages
produced by Micro-
Measurements
Residual Stress (Strain)
Gages produced by
Micro-Measurements
References: (1) ASTM E837: “Standard
Test Method
for Determining Residual
Stresses by the Hole-
Drilling Strain-Gage
Method”
(2) M-M TN-503:
“Measurement of Residual
Stresses by the Hole
Drilling Stain Gage
Method”
Alignment and Hole-Drilling Setup for
Residual Stress/Strain Measurement
(Former) ME556 Lab Experiment:
Residual Stress Measurement
Overall View of Set-up: Welded Hot Rolled Mild Steel Plate (ASTM A36), Residual Strain Gage Rosettes, P-3500 Strain Gage Amplifier, SB-10 Ten-Channel Switch and Balance Unit
Specimen : Welded Hot Rolled Steel Plate (Intended Rosette Pattern)
Welding
12 inch
24 inch
4 “
4 “
Rosette Bonding Line
0.75“
2“
2“2“
Symmetric Position
1.25“
#1 #2 #3 #4 #5 #6 #7 #8
#Center
Upper
Lower
Rosettes:Before (on Right)
andAfter (on Left)
Center Hole is Drilled
Definition of gage element 1, 2, and 3
123Element number: