Measurement of Residual Stress/Strain

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


(a) Stress transformation equations…label the new axes “r” and “θ ”(instead of x' and y')


θθτθσθσσ

θθτθσθσσ

θ

θ22

22

22

sincossincos

sincos2cossin

sincos2sincos

−+−=

−+=

++=

xyxxyyr

xyyyxx

xyyyxxr


(b) Stresses in a Uniformly-Loaded Thin Plate


(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


(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


(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


(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


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


(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


(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





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





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 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 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 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 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 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:

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Measurement of Residual Stress/Strain