Residual stresses in orthotropic materials

Residual stresses in orthotropic materials

Citation for published version (APA):

Gerwen, van, L. (1988). Residual stresses in orthotropic materials. (DCT rapporten; Vol. 1988.075). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1988

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RESIDUAL STRESSES IN ORTHOTROPIC MATERIALS (September

-

November 1988)

by

L. v. Gerwen

1

I

RESIDUAL STRESSES IN ORTHOTROPIC MATERIALS

Summarv

A first step has been made to find a method for determining residual

stresses in orthotropic materials. This method is based on the hole drilling strain gage method for isotropic materials as described in the

ASTM standard E837-85 (see Al 1-5). The step from isotropic to

orthotropic materials has been made possible by the use of a finite element analysis. This analysis can be done for each material, in which the residual stresses have to be determined. The results of the finite element analysis describe the relation between the measured strain relaxation and the residual stresses.

2

C O N T E N T S

1. INTRODUCTION

2. THEORY FOR ISOTROPIC MATERIALS

2.1 Distribution of stresses before drilling

2.2 Distribution of stresses after drilling

2.3 Formulas for residual stresses

3. ORTHOTROPIC MATERIALS

3.1 Stress-strain relations 3.2 Stress distribution 3.3 Finite element analysis

4. EXPERIMENTS

4.1 Strain as a function of û

4.2 Parameters as a function of the elastic constants

4.3 Practical test with a glassepoxy

5. EVALUATION OF THE PRACTICAL TEST 6. CONCLUSIONS

REFERENCES

TABLES AND FIGURES

APPENDIX

1:

ASTM E 837-85 3 4 4 5 7 7 7 8 10 10 1 0

11

12 13 14 TF4.1 1-5 TF4.2 1-25 TF4.3 1-4 Al 1-5

APPENDIX 2: Distribution of stresses in isotropic A2 1-3

materials after drilling a hole

APPENDIX 3: Integrating effect A3 1

APPENDIX 4: A method for measuring planar residual A4 1-10

stresses in rectangularly orthotropic mat er ials

APPENDIX 5: Test

APPENDIX 6: Incremental Residual Stress APPENDIX 7: Applied stress

A5 1-2 A6 1-2

A7

1

APPENDIX 8 : Procedure t o calculate residual stresses

3

1

INTRODUCTION

Residual stresses were often neglected, because it was not possible to determine their magnitudes and directions. However they can be of great influence in the failure behaviour of a material. For isotropic

materials different methods have been found to determine the residual stresses. The practical most used one is the hole drilling, strain gage method according to ASTM standard E837-85. This method is

semidestructive,which is a great advantage.

particularly fibre reinforced composite materials. The hole drilling method cannot be used, for these materials are not isotropic. In the isotropic case analytical formulas have been derived to describe the relation between the residual stresses and the relieved strain in radial direction after drilling a hole. Would it be possible to find such

formulas for more complicated materials?

Nowadays many engineers work with more complicated materials,

2 THEORY FOR ISOTROPIC MATERIALS

In this chapter the existing theory, on which the hole drilling method

is based, will be repeated. This will be done in the perspective of

expanding the theory, which was derived for isotropic material, to

orthotropic material. In 2.1 an expression will be found for the

distribution of stresses before drilling a hole. In 2.2 the same will be

done for the situation after drilling a hole. In 2.3 the final formulas

will be derived for the residual stresses.

As shown in fig.1 the situation is two dimensional, because there is a

free surface. The stresses al, a2, and a12 are regarded as uniform.

I

Figure 1 . Schematic d i a g r a m showing coordinote system, stress components, h o l e geometry. a n d s t r a i n - g a g e locations,

4

2.1 DISTRIBUTION O F STRESSES BEFORE DRILLING

First, for simplicity, a case is considered, where only is stress in one direction, as shown in fig.2:

fig.2

The distribution of stresses before drilling will be as follows:

orl =a1 cos2 8 ugl = o1 sin2 8

I

rTo1= - - sili 20 2

where '6 is the angle, which the examined point makes with the x-axis.

2.2 DISTRIBUTION O F STRESSES AFTER DRILLING

Next, when a circular hole of diameter d is bored, the distribution of

stresses (according to Flügge, Handbook of engineering mechanics, see

summary in A2 1-3) will be as follows (fig.3):

a r H = 5

(1

-$)

+$

(1

+r4

- ~ cos 28

2 3104

i

(I -

%+%)

sin 20.

5 r o H =

-'1

5

2.3 FORMULAS FOR RESIDUAL STRESSES

Hence the distributed stresses due to the drilling, which are

differences of the stress states expressed in eqs.(l) and eqs.(2) are as

follows :

Using Hooke's law the expression for the relieved strain in the radial direction becomes:

6 1 4p2 r 2 cos26,

I

This equation can be expressed generally as:

where

When there are stresses in two directions together with shear the

expression for the relieved strain becomes (see A4 4):

er= al(A+Bcos26)

+

a2(A-Bcos26)

+

012(Csin26)

As shown in fig.4

with their centers at the radius R from the center of the hole site and

located at the angles:

6

8 3=j3

e

*=j3+45 0

e

1=j3+90 0

The reference axes can be rotated until shear has become zero.In these

directions the stresses become u and u Generally, the directions of

the principal stress axes are no! known. Let /3 be the unknown angle.

Substituting the three angles in:

9'

E ~ = U (A+BcosZB)

+

CY (A-Bcos~B)

P q

and j3 for the measured

P' uq

produces three simultaneous equations in o

strains

€1,

€ 2 and € 3 . Solving for the principal stresses yields:

€ 3 ( A + B c o s ~ ~ ) - E ~ ( A - B c o s ~ ~ ) Dp = uq = 4AB cos 2 8 €1 ( A + B C O S 28) - €3 ( A - B C O S 28) 4AB cos 2 8 € 3 - 2 € 2

+

€1 tan 28 = €3 - € i

Because the practical strain gages are not points, but surfaces, the

corrected A and B values can be derived from an integration procedure

(see A3).

7

3 ORTHOTROPIC MATERIALS

In 3.1 a description will be given of orthotropic materials. In 3.2 it

will be seen, that it is not possible to derive analytical formulas for

the residual stresses in orthotropic materials. In 3.3 a solution will

be found in a finite element analysis.

3.1 STRESS-STRAIN RELATIONS

In most practical applications the material symmetry directions are

known and the associated elastic constants are known or can be determined experimentally. These elastic constants appear in the

following stress- strain equations relating the surface strains in the

material symmetry directions ( 1 , 2 ) to the corresponding stress

components:

Ell =engineering modulus in direction 1

E22 =engineering modulus in direction 2 NU12=Poisson's ratio

all =stress in direction 1

a22 =stress in direction 2 ~ i= i ( g i i / E i i ) - (vzi/Ez,)w22r

E22 = - ( v 1 2 / ~ 1 , ) u I ,

+

CT^./&^), G =shear modulus El:! = C T I * / c ' .

NU21=E22/Ell*NU12 al2 =shear

3.2 STRESS DISTRIBUTION

In the isotropic situation, after drilling a hole, a differential

equation for the stress function Q can be derived (see A2 1-3):

For orthotropic material such a differential equation for the stress function cannot be derived.

For isotropic materials the differential equation could be solved, because of the assumption that the stress distribution consists of two

parts. One part independent of û , the other part proportional to cos2û.

This assumption doesn't appear to be true for orthotropic materials (chapter4.1). This means, that, even if there was an ordinary

differential equation, it couldn't have been solved.

In 'A Method for Measuring Planar Residual Stresses in Rectangularly

Orthotropic Materials' (see A4 1-10), a solution for determining

residual stresses is based on the assumption that:

without any further explanation.These equations were true in the isotropic situation, because the distribution of stresses was

8

proportional to cos28. These equations aren't true in orthotropic cases.

This has been tested with a finite element analysis, with only a l l :

Ec=(A-B)all

The results from this test are differences of more than 40 percent for

eC after determining A and B from ea and Eb (see TF4.1 i).

Anifac=Ell/E22

eC1: Resulting from finite element analysis

~ ~ Resulting from assumption 2 :

Anifac E C 1 (10-10) rC2 difference 1.00 1.89 3.77 7.55 11.32 -1.0016 -5.77828 -4.47848 -3.46188 -2.94761

-.

9907 1 percent 34 percent -7.7437 -6.48015 45 percent -4.430442 30 percent -3.279974 11 percent 9 9 G=27. 27*109 NU=O. 32

where anifac= 1.00: E=72*10

9

anifac= 1.89 : Ell=23. 4*109 E22=12.4*10 G12=4.65*10 NU12=0.266

anifac= 3.77: E11=46.8*10 anifac= 7.55: E11=93.6*10 11 anifac=11.32: Ell=140.4*10 11 9 11 11 11 9 11 I1 11 11 9

3.3 FINITE ELEMENT ANALYSIS

The problem of determining residual stresses in orthotropic materials cannot be solved analytically. An other possibility is a finite element

analysis. The measured strain with the hole drilling method for

orthotropic materials will always satisfy:

~3=Gal

+

Hg2

+

1012

To determine these nine parameters, a finite element method can be used.

Because the superposition principle can be used the analysis can be made

for each stress situation separately. For example with only a l ,

parameters A, D and G can be determined.

The finite element analysis (fig.5) is very cheap. The calculation of 9 parameters costs about 10 guilders. The program to solve the

parameters and the program to calculate residual stresses from measured strain are described in appendix 8 (A8 1-12).

9

fig.5: 2D Finite element mesh to determine strain around the drilled hole for orthotropic materials.

10

4 EXPERIMENTS

4.1 STRAIN AS A FUNCTION OF 6

A finite element analysis has been done for four different orthotropic

cases with a uniform stress in one direction:

1 anifac=l. 89

2 anifac=3.77

3 anifac=7.55

4 anifac=ll. 32

For the results of this test see TF4.1 1-5. The conclusion from this

test is, that the relieved strain in radial direction is not a simple

function with respect to 6. The strain is not proportional to cos26.

4.2 PARAMETERS AS A FUNCTION OF THE ELASTIC CONSTANTS

So far, a method has been found to solve the nine parameters, which

describe strain as a function of stresses. This method has one

disadvantage. The parameters have to be determined for each material, in

which residual stresses have to be known. When the parameters could be

written as a function of the elastic constants, the method would be generalized for each orthotropic material. Seeing that this cannot be done analytically, an effort will be made with a finite element

analysis. In the model was started with the following elastic constants: Ell=50 GPa

E22=10 GPa G12= 5 GPa NU12=0.25

The stress situation was crl=-lOO N/mm2, 02=012=0 N/mm2.

Each elastic constant has been changed separately in order to find a relation between these constants and the parameters.The changes were as follows : Ell= 30 E22= 5 G12= 2 NU12= 0.1 40 45 7.5 10 3 4

-

50 12.5

-

5 55 15 6 0.15 0.2 0.25 0.3 60 70 17.5 20 80 25 7 8 9 0.35 0 . 4 0.45

11

After each change the relieved strain was determined. The results can be

seen in TF4.2 1-25.

Take for example:

E 1=Aa1

where Ell, E22, G12, and Nu12 are the underlined values. After changing the value of Ell, the strain will be changed too. The strain as a

function of Ell (TF4.2 4) looks like a second-degree polynomial of

l/Ell. It seems to be possible to estimate the parameter A as a function of Ell by regression with 1/Ell and 1/E112. After this regression the residuals appear to be negligible. This results in (see TF4.2 2-4,7):

E~=(AI

+

A2/Ell

+

A3/E1l2)a1

where E22, G12, and NU12 are constant. The same procedure can be followed for E22 and G12. The strain as a function of NU12 (TF4.2 22) shows a linear curve. This leads to (see TF4.2 8-10,13 TF4.2 14-16,19 TF4.2 20-22,25):

€1=(A4

+

A5/E22

+

A6/E222)ol Ell, G12, NU12 constant

~1=(A7

+

A8/G12

+

A9/G122)al Ell, E22, NU12 constant

E~=(AIO

+

AllNU12)al Ell, E22, G12 constant

Including all correlation effects ~1=Aal can be described as a function

of the elastic constants with 3*3*3*2=54 constants. After varying the hole diameter, the set of constants would be much larger. From this can be concluded, that the effort to work with formulas, which have been generalized for the elastic constants, requires a very large set of constants. These constants should be derived by a large set of finite element calculations. It is more practical to do a single finite element calculation tailored for the material tested, using the correct

(measured) hole diameter.

4.3 PRACTICAL TEST WITH A GLASSEPOXY

The finite element model has been tested for an orthotropic material. The material was a glassepoxy with the following estimated elastic constants:

E11=46.8 GPa E22=12.4 GPa G12=4.65 GPa NU12=0.266 GPa

In this test (see A5 1-2) a known stress has been tried to determine by drilling a hole in the material. With the finite element method the parameters could be derived to determine the known stress from the measured strains. This stress has been applied by bending the material. Each time the strains have been measured in the situation with and without bending. The incremental method was used (see A6 1-2). To get the same stress each time, an extra strain gage has been applied as a reference. From the difference between the measured strains with and

without bending, after correcting with the right k value, the applied

12

Known stress (see A7) :

al= 1 4 . 2 N/m2 U Z = 0 . 9 N/m2 012" 0.02 N/m2 Measured stress: U I = 17 N/m2 ~ 2 = 3 N/m2 O N / w 2

5 EVALUATION O F THE PRACTICAL TEST

The results of the test aren't very good due to several causes. The

finite element model is a 2D plain stress model and represents a hole

through the thickness method. This can only be done for plates, which

have a thickness smaller than the diameter of the hole. In the test the

thickness was about 4 mm. The diameter 2 mm.

An other disadvantage of the material was, that the constants weren't exactly known, but have been estimated.

The stress was applied by bending, which means that the stress wasn't uniform through the thickness. That's why the incremental method has been made use of. This method is based on the standard curve of

appendix 6 (see A6). This curve might not be correct for orthotropic

materials. The test would have been better, when the stress was applied by pulling. Then the stress would have been uniform through the

thickness.

The finite element model is two dimensional. In the isotropic situation

the three dimensional effects could be neglected. Probably this cannot be done in the orthotropic situation. The results would have been better

if the model had been three dimensional.

From these facts can be concluded that this test is not reliable. The test would be much better if:

-The material constants were better known. The finite element calculation has been done with estimated elastic constants.

-The thickness was 2 mm or the model was three dimensional.

13

6 CONCLUSIONS

1 Determining the stress strain relations for the hole drilling method to calculate residual stresses in orthotropic materials could not be done analytically.

2 A solution has been developed with the help of a 2D finite element

analysis.

3 The model could be improved by making it three dimensional.

4 It is not meaningful to generalize formulas for any possible material

It is better to use the finite element model tailored for the material, in which the residual stresses have to be determined.

5 When sticking to a 2D analysis the finite element analysis should be

done in the best way with a wellknown material with a thickness of

14

REFERENCES

-

ASTM E 837-85: Standard test method for determining residual stresses

by the hole drilling strain gage method.

-

COMPOSITE MATERIALS, Vol. 2 , No. 2 , p. 244: A method for measuring

planar residual stresses in rectangularly orthotropic materials.

-

Sci. Papers I.P.C.R., Vol. 59, No. 2 , p.69: A method for measuring

stresses on metal surface photoelastically.

-

Measurements Group, Tech Note 503: Measurement of residual stresses by

the blind hole drilling method.

- Centre for Composite Materials, Imperial College of Science &

Technology, London: Introduction to fibre reinforced composites.

-

Handbook of engineering mechanics, Flügge.

TF4.l

1

Relation between strain and angle. -

OBS ANGLE STRAIN

1 90 -5.778283-10 2 75 -4.794543-10 3 60 -1.856083-10 4 45 2.471153-10 5 30 7.620373-10 6 15 1.168893-09 7 O 1.26860E-09

OBS ANGLE STRAIN

8 90 -4.478483-10 9 75 -3.817153-10 10 60 -1.836063-10 11 45 1.129253-10 12 30 5.295463-10 13 15 9.242453-10 14 O 8.738653-10

OBS ANGLE STRAIN

15 90 -3.461883-10 16 75 -3.008513-10 17 60 -1.624423-10 18 45 4.660943-11 19 30 6.880033-10 20 15 1.407403-09 21 O 5.36263E-10

OBS ANGLE S TRAIN

22 90 -2.947613-10 23 75 -2.592793-10 24 60 -1.48230E-10 25 45 2.315483-11 26 30 2.840493-10 27 15 7.07924E-10 28 O 3.743073-10

TF4.1 2

(d

+

+

+

+

+

+

I

*E

'rn

O b O (D O u7 W -I e Z a O P O o o (u O rl O 11111111111111111111 z~mmmooooooooo~oooooo ~IIIIIIIIIIIIIIIIIIII rl~rl-imm~~0tnvmcu-itn~~m~u7~0 HQQQQrlrlrlrlrl-irl-i-iddrldrlrlrl Kw~wwwwwwwwwwwwwwwwww l-mN~ooooooooooooooooo v)

...

IIIIII

TF4.1 3

a,

d

cd

‘ij

G

cd

+

+

+

a O P O m

+

+

O m O m O b O co O in W O d O

TF4.1 4

.

a,

d

a,

s

G

O

cd

a,

P=

&

+

+

+

+

S T R A I N 8.OE-10 7.OE-10 6.OE-10 5 . O E - 1 0 4.OE-10 3.OE-10 2.OE-10 1 . OE-IO 2.5E-17 -1 .OE-10 -2.OE-10 -3.OE-10 t

Relation between

strain

and angle.

A N I F A C - 1 1 . 3 2 2 5 8

+

+

+

+

I

+

₊

I O 10 20 30 40 50 60 70 80 90 I I I _I I I I I I I I I I I I + i A N G L E

Tf4.2 1

TF4.2 2

OBS ANGLE STRAIN

1 O 1 . 1 7 5 2 8 E - 0 9 2 45 1 . 8 3 8 8 2 3 - 1 0 3 90 - 6 . 3 1 4 6 5 3 - 1 0 4 O 9 . 9 3 9 3 1 3 - 1 0 5 45 1 . 2 6 9 1 8 3 - 1 0 6 90 - 5 . 6 5 9 8 7 3 - 1 0

OBS ANGLE STRAIN

7 O 9 . 2 3 9 2 0 3 - 1 0 8 4 5 1 . 0 7 6 1 2 3 - 1 0 9 90 - 5 . 4 1 2 8 5 E - 1 0 OB S ANGLE STRAIN 10 O 8 . 6 3 4 0 3 3 - 1 0 11 4 5 9 . 2 0 6 7 7 E - 1 1 1 2 90 - 5 . 2 0 0 9 4 3 - 1 0 OB S ANGLE STRAIN 1 3 O 8 . 1 0 4 9 6 3 - 1 0 1 4 45 7 . 9 2 7 5 8 3 - 1 1 1 5 90 - 5 . 0 1 5 9 6 3 - 1 0

OBS ANGLE STRAIN

1 6 O 7 . 6 3 6 8 6 3 - 1 0 1 7 4 5 6 . 8 5 7 0 7 3 - 1 1 1 8 90 - 4 . 8 5 2 2 0 3 - 1 0

714.2 3

Strain as a function o f ~1-e~~)

OBS ANGLE STRAIN

1 9 O 6 . 8 4 3 4 5 3 - 1 0 20 4 5 5 . 1 6 2 7 5 3 - 1 1 2 1 90 - 4 . 5 7 3 1 9 E - 1 0

OBS ANGLE STRAIN

22 O 6 . 1 9 3 6 5 3 - 1 0

23 4 5 3 . 8 8 0 3 5 3 - 1 1

Strain

as

a function

of

E l

1

Direction of fibres (angle=O")

STRAIN 1.2E-09 1.1E-OE) 1 . O E - 0 9 9 . O E - 1 0 8.OE-1C 7 , O E - 1 C 6.OE-1C P

TF4.2

5

TF4.2 6 O 4

4

4

W

O

eb.i n

do

+

W O a, O

+

W O b 4 O

+

W O u3 4 O

+*

w* ow m 4 O

+

W O P 4 O

+

W O m rl O

i

rl

TF4.2 7

Residuals after regression with 1/Ell and . 1/311/311

OBS Ell STRAIN RESID

1 30000000000 2 40000000000 3 45000000000 4 50000000000 5 55000000000 6 60000000000 7 70000000000 8 80000000000 1.175283-09 9.939313-10 9.239203-10 8.634033-10 8.104963-10 7.636863-10 6.843453-10 6.193653-10 9.613383-13 -3.00325E-12 -9.821353-13 7.70137E-13 1.883243-12 2.19984E-12 7.946193-13 -2.623793-12

- - - _ - - - -Direction makes an angle of 45" with f i b r e s . - - -

OBS Ell STRAIN RESID

9 10 11 12 13 14 15 16 OBS 17 18 19 20 21 22 23 24 30000000000 40000000000 45000000000 50000000000 55000000000 6 O O O O OB O O O O 70000000000 80000000000 1.838823-10 4.680903-15 1.269183-10 -7.888303-15 1.076123-10 -1.286463-14 9.206773-11 -3.442983-15 7.927583-11 6.079663-15 6.857073-11 2.062153-14 5.162753-11 1.764753-14 3.880353-11 -2.483373-14

- - - -Direction fibres (angle=90")

---'----

Ell 30000000000 40000000000 45000000000 50000000000 55000000000 60000000000 70000000000 80000000000 STRAIN RESID -6.314653-10 -4.288363-13 -5.659873-10 1.315223-12 -5.412853-10 4.715103-13 -5.200943-10 -3.23065E-13 -5.015963-10 -8.362263-13 -4.852203-10 -1.007713-12 -4.573193-10 -4.016333-13 -4.341983-10 1.210743-12

OBS ANGLE STRAIN 1 O 9 . 2 3 9 4 3 3 - 1 0 2 45 - 1 . 9 9 9 3 0 3 - 1 1 3 90 - 7 . 4 4 1 8 2 3 - 1 0 4 O 8 . 8 6 1 5 7 3 - 1 0 5 45 5 . 2 8 6 5 7 3 - 1 1 6 90 - 6 . 0 2 5 1 6 3 - 1 0

OBS ANGLE STRAIN

7 O 8 . 6 3 4 0 3 3 - 1 0

8 45 9 . 2 0 6 7 7 3 - 1 1 9 90 - 5 . 2 0 0 9 4 3 - 1 0

OBS ANGLE STRAIN

10 O 8 . 4 7 8 5 6 3 - 1 0

11 45 1 . 1 7 0 1 7 3 - 1 0 1 2 90 - 4 . 6 4 0 2 5 3 - 1 0

OBS ANGLE STRAIN

1 3 O 8 . 3 6 3 7 1 3 - 1 0

14 45 1 . 3 4 4 1 4 3 - 1 0

15 90 - 4 . 2 2 2 6 4 3 - 1 0

OBS ANGLE STRAIN

1 6 O 8 . 2 7 4 7 2 3 - 1 0 1 7 45 1 . 4 7 3 6 0 3 - 1 0 1 8 90 - 3 . 8 9 3 1 5 3 - 1 0

TF4.2 9

OBS ANGLE STRAIN

1 9 O 8 . 2 0 3 0 2 E - 1 0 20 45 1 . 5 7 3 2 2 E - 1 0 2 1 90 - 3 . 6 2 2 4 5 3 - 1 0

OBS ANGLE STRAIN

22 O 8 . 0 9 3 9 6 3 - 1 0

23 45 1 . 7 1 6 8 2 3 - 1 0 24 90 - 3 . 1 9 6 0 4 3 - 1 0

TF4.2 10 O rl

+w

2000000 0000000- ai I I 1 1 I 1

uwwwwwwwwwwwwww

I-~cu~o~~~~~PoN~o cn- mmmmmmmmmmmmmm I I I I I I I I i 'O HdrlrlrlrldZrlrlddddrlV I 1 I 1 I I 1

TF4.2

11

O rl Z0~~00~0~0rlrlrl~rlrlrl~~~~rl Mrlrlrlrlrlrlrl~rlmm~~~pmNrl~~~ I1 H e. r! e. d ddrlTIrlrlrlrlrlrlrlrlrl QllllllllltllllllllIll ~wwwwwwwwwwwwwwwwwwwww ~~~~~~mNrlU0000~000u000

...

O

+

W P N rl O

+

W rl m" rl N N W O rl m O

+

eu O m m U

+

W O T

TF4.2 12 O rl

02

02

L4

-"

111111111111111111111 I I 200000000000000000000000 uwwwwwwwwwwwww~wwwwww~ulw (n

...

rnrnrnrnrnPPPPP~~~~~~~~~~~~~ 111111111111111111111 I1 Hrlrlrlrl-44-4-4~~~rlrlrlrlrlrl~44~~~ ~IIIIIIIIII~IIIIIII~III~ ~rlrn~~mrl~~~mrlrn~~rn-4~~~~rl~~ N O

+

W P Cu 4 O

+

W -4 N N W

2

+

W P -4 m O

+

Li9 O

m

m

O

+

W O P

TF4.2

13

OBS E22 STRAIN RESID

5000000000 7500000000 10000000000 12500000000 15000000000 17500000000 20000000000 25000000000 9.239433-10 3.319153-13 8.861573-10 -1.180613-12 8.634033-10 9.971693-16 8.478563-10 6.715803-13 8.363713-10 7.722753-13 8.274723-10 5.265053-13 8.203023-10 5.413573-14 8.093963-10 -1.176793-12 _ - - _ - - _ _ _ _ - _ _ _ _ - - - - D i r e c t i o n makes an angle of 45" w i t h f i b r e s . - - -

OBS ' 32 2 STRAIN RESID

9 10 11 12 13 14 15 16 5000000000 7500000000 10000000000 12500000000 15000000000 17500000000 20000000000 25000000000 -1.999303-11 5.286573-11 9.206773-11 1.170173-10 1.344143-10 1.473603-10 1.573223-10 1.716823-10 -2.065233-13 7.618723-13 -6.610983-14 -4.424103-13 -4.729663-13 -2.424783-13 3.677903-14 6.318353-13 17 1 8 19 20 21 22 23 24 5000000000 7500000000 10000000000 12500000000 15000000000 17500000000 20000000000 25000000000 -7.441823-10 -1.721103-12 -6.025163-10 6.00253E-12 -5.200943-10 2.752323-13 -4.640253-10 -3.321513-12 -4.222643-10 -4.161423-12 -3.893153-10 -3.024043-12 -3.622453-10 -5.702003-13 -3.19604E-10 6.52049E-12

TF4.2 14 OBS 1 2 3 ANGLE O 45 90 STRAIN 1 . 0 2 5 6 0 E - 0 9 1 . 7 6 1 6 4 3 - 1 0 - 5 . 5 9 8 4 9 3 - 1 0 4 5 6 O 45 90 9 . 6 9 8 7 4 3 - 1 0 1 . 3 5 4 2 8 E - 1 0 - 5 . 4 6 8 9 9 3 - 1 0 OBS 7 8 9 ANGLE O 45 90 STRAIN 9 . 1 2 6 8 6 3 - 1 0 1 . 0 9 6 1 6 3 - 1 0 - 5 . 3 2 6 1 0 3 - 1 0 OBS ANGLE 1 0 11 1 2 O 45 90 STRAIN 8 . 6 3 4 0 3 3 - 1 0 9 . 2 0 6 7 7 E - 1 1 - 5 . 2 0 0 9 4 3 - 1 0 OBS ANGLE 1 3 1 4 1 5 O 45 90 STRAIN 8.22069E-10 7 . 9 4 7 7 6 E - 1 1 - 5 . 0 9 5 5 4 3 - 1 0 OBS ANGLE 1 6 1 7 1 8 O 4 5 90 STRAIN 7.87233E-10 7 . 0 0 6 7 6 E - 1 1 - 5 . 0 0 6 8 6 E - 1 0

TF4.2 15 S t r a i n as a function o f G2@lPm!9 OBS ANGLE 1 9 20 2 1 O 45 90 STRAIN 7 . 5 7 6 1 6 E - 1 0 6 . 2 7 9 6 2 3 - 1 1 - 4 . 9 3 1 8 9 3 - 1 0 OBS ANGLE 2 2 23 24 O 45 90 STRAIN 7 . 3 2 1 7 0 E - 1 0 5 . 7 0 8 1 6 3 - 1 1 - 4 . 8 6 7 8 6 3 - 1 0

TF4.2 16 ö

%

z

H Q U k (B m m O

+

W O 00 m O

+

W O b w O w O Ln m O

+

W O m O m m 8

+

W O N

m O

+

W O N

t

m O

+

w O P

f

m O

+

w O m

rain

as

a

ion

of

G 1 2

Direction fibres (angle=9O0)

STRAIN

-4.8E- 110

1

STRAIN

-4.8E- 110 -4.9E-IO -5. OE-IO -5. I E - I O -5.2E-10 -5.3E-10 -5.4E-10 - 5 . 5 E - I O -5.6E-10

2.OE-k-09 3.OE+09 4.0E+09 5.OE-l-09 6.OE+09 7.OE+09 E3.OE-l-09 9.OE-l-09 612

TF4.2 19 1 2000000000 2 3000000000 3 4000000000 4 5000000000 5 6000000000 6 7000000000 7 8000000000 8 9000000000 1.02560E-09 1.968143-12 9.698743-10 -7.612683-12 9.126863-10 1.095073-12 8.634033-10 5.053943-12 8.22069E-10 4.925153-12 7.872333-10 2.317343-12 7.576163-10 -1.598163-12 7.321708-10 -6.148793-12

- - - _ _ _ - _ - - _ -Direction makes an angle of 45" with f i b r e s - - -

OBS G12 STRAIN RESID

9 2000000000 10 3000000000 11 4000000000 12 5000000000 13 6000000000 14 7000000000 15 8000000000 16 9000000000 1.761643-10 5.229103-14 1.354283-10 -2.304993-13 1.096163-10 1.094973-13 9.206773-11 1.476393-13 7.947763-11 7.216703-14 7.006763-11 -1.199003-14 6.279623-11 -7.727693-14 5.708163-11 -6.182753-14

OBS G12 STRAIN RESID

17 2000000000 18 3000000000 19 4000000000 20 5000000000 21 6000000000 22 7000000000 23 8000000000 24 9000000000 -5.598493-10 -5.468993-10 -5.326103-10 -5.200943-10 -5.095543-10 -5.006863-10 -4.931893-10 -4.867863-10 -4.852893-13 1.88913E-12 -3.026193-13 -1.257133-12 -1.188903-12 -5.323563-13 4.061993-13 1.470963-12

TF4.2 20 S t r a i n as a function o f - K l 2

OBS ANGLE STRAIN

1 O 8.67434E-10 2 45 6 . 7 8 1 2 5 E - 1 1 3 90 - 5 . 4 7 2 3 9 E - 1 0 4 O 8 . 6 6 1 0 1 E - 1 0 5 45 7 . 5 8 9 7 6 E - 1 1 6 90 - 5 . 3 8 2 3 9 3 - 1 0

OBS ANGLE STRAIN

7 O 8 . 6 4 7 9 9 3 - 1 0

8 45 8 . 3 9 8 7 1 3 - 1 1 9 90 - 5 . 2 9 1 8 9 3 - 1 0

OBS ANGLE STRAIN

10 O 8 . 6 3 4 0 3 E - 1 0

11 4 5 9 . 2 0 6 7 7 3 - 1 1 1 2 90 - 5 . 2 0 0 9 4 E - 1 0

OBS ANGLE STRAIN

1 3 O 8 . 6 2 0 9 4 3 - 1 0 14 45 1 . 0 0 1 4 8 E - 1 0 1 5 90 - 5 . 1 0 9 5 6 3 - 1 0

OBS ANGLE STRAIN

1 6 O 1 7 45 18 90 8 . 6 0 6 9 8 3 - 1 0 1 . 0 8 2 3 3 3 - 1 0 - 5 . 0 1 7 6 7 3 - 1 0

TF4.2 21

Strain as a function o f NU12 _ .

OBS ANGLE STRAIN

1 9 O 8 . 5 9 3 9 0 E - 1 0 20 45 1 . 1 6 3 1 0 E - 1 0 2 1 90 - 4 . 9 2 5 3 5 E - 1 0

OBS ANGLE STRAIN

22 O 8 . 5 7 9 8 7 3 - 1 0

23 4 5 1 . 2 4 3 9 5 3 - 1 0 24 90 - 4 . 8 3 2 5 8 3 - 1 0

rain

as

a

unction

of

NU12

D i r e c t i o n o f f i b r e s (angle=O") STRAXN 8.7E- 110 8.7E-$0 8.7E-10 8.6E-iO 8.6E-10 8.6E-iO 8.6E-i0 8.6E-10 8.6E-10 8.6E-10 8.6E-10 8.6E-10 0.09 0.14 0.19 0.24 0.29 NU12 0.34 0.39

o

,44

o

,49-

114.2 23 I I I I I I 1 r m .P O P .- O m .m O P .m O ma

*

-3

oz

P -a O m O il P 0 il 0-I -0 rl ri rt do il rl il .d O I O I W i! W I O w I ri O O O I O W [Iw W i-m (u rl rl rl m m b u) O I W sl 4. I W

EZ

ai

TF4.2 24 O O O I O w I w I 4 N P LD Cr) W

1

w m m Q lil lil LD I u3 I Ln I I

1

1

P 4 O I w rl rl O I

w

lil I sd si O O 1 I W w 0 r! e. c=: I

TF4.2 25 Residuals a f t e r r e g r e s s i o n w i t h NU12 OBS Nu12 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 O . 30 0 . 3 5 0 . 4 0 0 . 4 5 STRAIN 8 . 6 7 4 3 4 E - 1 0 8 . 6 6 1 0 1 3 - 1 0 8 . 6 4 7 9 9 3 - 1 0 8 . 6 3 4 0 3 E - 1 0 8 . 6 2 0 9 4 E - 1 0 8 . 6 0 6 9 8 3 - 1 0 8 . 5 9 3 9 0 3 - 1 0 8 . 5 7 9 8 7 3 - 1 0 RESID - 2 . 4 7 4 1 4 3 - 1 4 - 9 . 1 4 3 8 0 3 - 1 5 3 . 7 4 5 6 3 3 - 1 4 - 1 . 0 0 6 8 2 3 - 1 4 3 . 0 5 2 7 0 E - 1 4 - 1 . 6 9 9 7 5 3 - 1 4 2 . 3 4 5 8 0 3 - 1 4 - 3 . 0 4 9 0 4 E - 1 4 D i r e c t i o n makes an a n g l e o f 45" w i t h f i b r e s - - - - _ _ - - _ _ - - _ _ _ _ _ _ _ _ _ OBS NU12 9 0 . 1 0 10 0 . 1 5 11 0 . 2 0 1 2 0 . 2 5 1 3 0 . 3 0 1 4 0 . 3 5 1 5 0 . 4 0 1 6 0 . 4 5 STRAIN 6 . 7 8 1 2 5 3 - 1 1 7 . 5 8 9 7 6 3 - 1 1 8 . 3 9 8 7 1 3 - 1 1 9 . 2 0 6 7 7 3 - 1 1 1 . 0 0 1 4 8 E - 1 0 1 . 0 8 2 3 3 3 - 1 0 1 . 1 6 3 1 0 E - 1 0 1 . 2 4 3 9 5 3 - 1 0 OBS 1 7 1 8 1 9 20 2 1 22 23 24 Nu12 RESID - 4 . 0 2 6 5 9 3 - 1 5 - 1 . 7 3 8 9 8 E - 1 5 4 . 8 7 7 8 0 3 - 1 5 2 . 6 9 6 5 9 3 - 1 5 5.15377E-16 2 . 8 0 2 9 9 3 - 1 5 - 3 . 7 0 7 3 9 E - 1 5 - 1 . 4 1 9 7 8 3 - 1 5 STRAIN 0 . 1 0 - 5 472393-10 0 . 1 5 - 5 . 3 8 2 3 9 3 - 1 0 0 . 2 0 - 5 . 2 9 1 8 9 3 - 1 0 0 . 2 5 - 5 . 2 0 0 9 4 3 - 1 0 0 . 3 0 - 5 . 1 0 9 5 6 3 - 1 0 0 . 3 5 - 5 . 0 1 7 6 7 3 - 1 0 0 . 4 0 - 4 . 9 2 5 3 5 3 - 1 0 0 . 4 5 - 4 . 8 3 2 5 8 3 - 1 0 RESID 1 . 6 1 9 3 7 3 - 1 3 2 . 1 5 7 9 4 3 - 1 4 - 6 . 8 5 0 3 7 3 - 1 4 - 1 . 1 4 5 9 7 3 - 1 3 - 1 . 1 6 5 6 0 3 - 1 3 - 6 . 8 2 4 9 3 3 - 1 4 2 . 4 0 5 1 5 3 - 1 4 1 . 6 0 3 4 2 3 - 1 3

TF4.3

1

OBS 1 2 3 4 5 6 7 8 9 10 11 12 13 1 4 15 16

HOLEDRILLING METHOD: MEASURED STRAIN

Depth in microinches STRESS without with with without without with with without without with with without without with with without DEPTH O O 10 10 20 20 30 30 40 40 50 50 60 6 0 7 0 7 0 EPS1 O 302 237

-

19 - 8 220 186

-

18 - 13 192 184

-

16 - 15 189 178 - 16 EPS2 O 149 131 O O 127 112 -1 -1 112 107 O -1 108 104 O EPS 3 O - 8 O 6 6 O O O 4 4 8 1 4 9 11 O EPSREF 0 327 328 0 0 328 328 1 1 324 3 24 1 1 327 327 1

TF4.3 2

ManiuulatinP data

cl= Strain in direction of fibres

c2= Strain in direction that makes an angle of 45" with fibres

~ 3 = Strain in direction i- fibres

et=&€

*

100/perc.

*

k1/k2 with kl=2.00 and k2=2.03 depth DO O 0.14 0.28 0.42 0.56 0.70 0.84 0.98 1.Incremental method

€1

€2 €3 302 149 -8 256 131 -6 228 127 -6 204 113 O 205 113 O 200 107 7 204 109 5 194 104 11 6 6 1 -46 -2 8 -24 1 -5 4 -10 662 -1 8 -4 -14 O -6 2 -5 SE3 2 O 6 O 7 -2 6 perc. 17 21 21 16 12 8 4

2.Hole through the thickness method

€lt €2t -267 -104 -131 -19 -113 -66 6 O -41 -49 49 25 -246 -123 €3t 12 O 28 O 57 -2 5 148

This method is not correct because the tension is not uniform through

the thickness and the thickness is greater than the diameter of the

drilled hole. The result is only used as an indication.

€1

€2 € 3 6 6 1 662 663 perc. Elt ~2~ ~3~

TF4.3 3 OBS ANGLE O 45 90 O 45 90 O 45 90 PARAMETERS Depth in microinches Stress in N/mm/mm SIG1 - 10 - 10 - 10 O O O O O O SIG2 O O O - 1 0 - 1 0 - 10 O O O SIG12 O O O O O O 10 10 10 PAR -0.000006596 - 0 . 0 0 0 0 0 1 7 3 4 0 . 0 0 0 0 0 3 6 3 1 0.000003039 - 0 . 0 0 0 0 0 4 9 1 6 -0.000014105 0.000000003 0.000000229 - 0 . 0 0 0 0 1 4 9 9 8

OBS DEPTH t o t a l incrl incr2 incr3 incr4 incr5 incr6 incr7 EPSOO -0.000100 -0.000267 -0.000131 -0.000113 O. 000006 O. 000049 -0.000041 -0.000246 RESIDUAL STRESSES EPS45 -0.000044 -0.000104 -0.000019 -0.000066

o.

000000 -0.000049 O. 000025 -0.000123 EPS90 O . 000015

o.

000012

o.

000000 O. 000028

o.

000000 O. 000057 -0.000025 O. 000148 SIGMA1 16.6444 45.4598 22.4958 18.4150 4.9813 36.8782 -1.0297 -7.5105 SIGMA2 3.2203 10.8122 5.7230 2.7842 -0.2607 -2.6831 -0.1767 -0.9095 SIGMA12 -0.0462 -1.8656 -3.2099 1.3589 O. 2045 3.5706 4.2355 -0.7407

A i i

APPENDIX

1

This standard is issued under the fixed designation E 837; the number immediately followieg the designation indicates the year of

original adoption or, in the ose of revision, the year of last revhion. A number in parenthcses indicates the year of last reapproval. A

s u p e e p t epsilon (e) indicates an editorial change since the last revision or reapproval.

INTRODUCTION

The hole-drilling strain near the surface of a mat

hole in the vicinity of the gages, and measuring the relieved s related to relieved principal stresses through a series of equations.

method is a semidestructive method for measuring residual stresses The method involves attaching strain gages to the surface, drilling a

s. The measured strains are then

1.

scope

1.1

This test methoa covers the procedure for determining residual stresses near the surface of isotropic elastic materials. Although the concept is quite generai, the test method de-

scribed here is applicable in those cases where the stresses do

not vary significantly with depth and do not exceed one half of the yield strength. Since the method is semidestructive, it

should be applied in those instances where a s

hole will not affect the usefulness of the struuct

tageous technically or ?nomi- volve hazardous materials, oper- is standard does not purport to

address all of the safety problems associated with its use. It is the responsibility of whoever uses this standard to consult and

est ab1 ish appropri~te nd health practices and deter-

mine rhe a p p l ~ c ~ ~ i t y ory limitations prior to use.

2.1

ASTMStandard,

E 251 Test Methods €or Performance Characteristics of Strain Gages2

3.1 Strain gages, in the form of a three-element rosette, are placed in the area under consideration.

3.2 A hole is driiied in the vicinity of the strain mes to a depth slightly greater than its diameter.

3.2.1 The residual strews in the area surrounding the driiled hole relax. The relieved strains are measured with a suitable strain-recording instrument. Within the close vicinity of the hole, the relief is nearly mmplette when the 0ept.h of the drilled hole approaches 1.2 times the diameter.

3.3 The surface strains relieved are related to the relieved principal stresses by the following relationship:

tr = ( A

+

B COS 2a)ux

+

( A - B COS 2a)uY

’

This tesf method i% under the jurisdibion of ASTM Committee E28 on

TAe;; & +& -&id:+., U”‘., V. .4 z g J 3

6, 1985. Pubiished June 1985. Y

dîibnE837-81.

, Vols 03.01 and 12.02.

where:

er = radial strain relieved at point P, l + p

1

A = - -

2E

x F y

uX, uY =principal stresses present in the structure before

drilling,

(Y

E = Young’s modulus,

p = Poisson’s ratio, = - (see Fig. i),

r

D = diameter of gage circle, and

Do = diameter of drilled hole.

3.3.1 Measuging the relieve8 radial strains ei, c2, and e3 at

points Pi, P2, and P3 (Fig. i), respectively, provides sufficient

inf~rmation to calculate the principal stresses u, and uv and

their orientation, 8, with respect to an arbitrarily selected reference.

3.3.1.1 It is assumed that the variation of the stresses in

the x and y directions is small and the variation with depth

is negligible.

4. SigniZiaance a d Use

= angie between the directions oft, and gx,

D DO

4.1 Residual stresses are present in almost all structures.

fla.

1

70 1

7 / U / \ / \

\

1 ' FIG. 2 T

They may be present as a result of manufacturing processes or they may occur during the life of the structure. In many cases residual stresses are a major factor in the failure of a structure, particularly one subjected to alternating service loads or corrosive environments. Residual stress may also be beneficial as, for example, compressive stresses produced by shot peening. The holedrilling strain-gage technique is a

practical semidestructive method for determining residual stresses.

5.1.1 The gages shall be oriented in a radial k t i o n , center of the rosette.

shall be in mutually perpendicular di- rections, with-a third along one of the bisectors (see Fig. 2).

NOTE I-This geometrical configuration was proposed by Rendier and Vigness It greatly simplifies the dadation ofthe residual

stresses?

5.1.2 The length and the width of the a d v e strain gage grid shall not exceed the diameter of the drilled hole.

5.1.3 The center of the gage circle shall be clearly identifi- able both before and &er the drilling operation.

5.1.4 The application of the stAn gage (cementing, wiring,

protective coating) shdl closely follow the manufacturer's recommendations, and shall ensure the protection of the strain gage grid during the drilling operation.

5.2 The three strain gages shall remain permanently con- nected and the stability of the installation shall be verified. The resistance to ground shail be no less than 20 o00 Ma.

' The boldface numben in parenthess refer u> the lin of referenas at the end

' A strain gage pettern of this is mufacliared bj. Measurrments Group,

of this test mahod.

Wendell, NC.

6 . IQS

6.1 The inmxhenta~on for recording of strains shall have a strain r&dution ofk2 X 1W6, and stability and repeatability

of the m ~ u ~ ~ e n t shall be at least I 2 x The lead

as short as practicable and a sating circuit should be used

or to cementing the strain endations of the manufac- turer of the cement used to attach the strain gage.

cleanhg and degreasing is required.

7.1.2 A smooth surface is usually necessary for strain gage application. However, abrading or grinding that could appre- ciably alter the s d a c e stresses must be avoided.

NOTE 2-In genemi, surface preparation should be restricted to those

methods which have,&n demonstrated to induce no significant residual

surface stresses.

8.1 Drilling:

8.1.1 Size of the drilled hole:

8.1,l.

1

Based upon prior experience, the diameter, Do, of

the drilled hole should be related to the diameter of the gage circle, D, by

D 2.5 <

-

< 3.4

Do

NOTE 3-As the mho DIDO hxase§, the sensitivity of the method

decreases. That is, the relieved strains are smaller.

8.1.1.2 The final depth of the hole shall be

1

.2D0 or, in the

case of sheet materials, the total thickness, whichever is smaller.

8.1.2 The center of the drilled hole shall coincide With the

center of the strain gage circle to within i0.015Do. Errors

AI 3 I Collet Flexiblo Coupling I Depth Control Guide Bor Otraln Gago Rosette

due to misalignment of the driiied hole could produce signif- icant errors in the calculated stress. To avoid these errors, it is recommended that an optical device be used for

the tool holder. A device suitable for this is shown

eliminate the introduction of plastic deformation in the surrounding the drilled hole.

8.1.3.1 Several driiüng techniques were investigated and reported to be suitable for the holeddihg method in tech- nical papets:

(a) End mi& carbide drills and modified end d s were 8.1.3 S e k t the driiiing operation and tool to minimize or

(b)

(4

Since any residual stress created by the selected drilling

methoù wiii adversely affect the accuracy of resdts, a verifi- cation of the seiected process is recommended when no prior

experience is available. Such verification could consist of applying a strain gage rosette, identical to the rosette used in the test, to a stress-free specimen of the same nominal com-

position, and then driiiing a hole. If the drilling method .is

satisfactory, the stpesses produced by drilling will be small. 8.1.3.2 When end mills are used, proceed with the drilling

using very light axid t h s t and slowly, to pernit ample time for heat dissipation.

8.1.4 CaKy out the hole drilling test at constant tempera- ture.

readings from each gage before starting

on and obtain strain readings cl, c2, c3

thereafter at intervals as the hole depth is increased incre-

mentally. The increments should be 10 % of the total depth, should yield a plot

wn in Fig. 4, with

for depth Do and

1

.XIo.

out of the scatter. band

the thickness, and the

numbers (I), (2), and-(3) in

principal stresses cx _and u, are located cs” clock-

wise from directions ( I ) and (3) when @ is positive. Compute the angie B from

“.’-

c3 - 2c2

+

t #

tan 28 =

9.1.3 Compute the stresses ux and u, fiom

€ 3

-

€ 1

ux,u =

*

f

J

J(tl

-

t2)2 4- (t2

-

€3)2

9.1.3.1 In the id&% case of a pint-sized strain gage

and a through-the-thickness drilled hole, the constants

2

and

E

are identical to A and B defined in 3.3. In order to account

for the integrating effect of a finite-sized gage, and for a blind hole situation, these constants must be computed from the

following equations:

A=-((1 +p)/2E)á

B

=

-

(1/2E) d

6

are dimensionless, material-independent cod-

by the equations in 9.1.3.2 ahd by Table

1.

See nt of Poison’s ratio and the NOTE 4-The codticient d

A i 4

7

Relieved Strain, % of strain relieved at depth = 1.2 Do

1 O0 80 60 40 20 0.2 0.4 0.6 0.8 1.0 1 Hole Depth Hole Dia. Do ____j) 2 Depth/Do

elieved Strain versus Depal for Stress Uniform Through the Thickness

coefficient b is nearly i 0.28 to 0.33, the values

a

and 6 are obtained by i gage area (9, 10).

For values of Poisson’s ratio from

Table

1

are correct to within i %.

9. I .3.2 In the case of a throuh-the-thickness drilled hole, ng the gage output over the

TABLE 1 Numerkai Vslws of Coeficienîs ü ami b

NmE-ï’he numerical values shown were computed for the gage geometry

d e f i n e d b y G L = G W , r , = 0 . 6 9 1 , r 2 = 1.31,andw=0.619.

Throu&-the-bi&nm hole” Blind hole, depth =z

r = DIDo DOB - - a 6 a d - sin 282~~sZe2 2r2’ - sin 2û2)

- -

where: - w = (2GW/D), rl = ( 2 R l / D ) , r2 = (2R2/D), and d

G W , D, R , , R z ,

e,,

and û2 are defined in Fig. 2 and r is as

defined previously. The numerical values computed from these equations are given in Table i for the most typical strain gage.

9.1.3.3 In the case of a blind hole with depth of

1

.2D0 the

values of a and b must be obtained from Table

1.

These numerical values were derived from the results of a finite elememt analysis (9) and were found to be in excellent agree-

ment with experimental results.

9. i .3.4 The constants A and relating the stresses u, and uy to the measured strains t, can be established using a

calibration experiment. The experimental calibration elim- inates errors due to the integration effects of the strain gage and to any imperfect geometry of the hole. Cement a rosette gage identical to the one used in the test to a calibration bar made of the same materiai as that ‘being invesrigated. The directions (3) and ( I ) of the gage should coincide with the

2.50 0.158 2.60 0.146 2.70 O. 136 2.80 0.126 2.90 0.118 3 . 0 0.110 3.10 0.103 3.20 0.0965 3.30 0.0907 3.40 0.0855 0.453 0.425 0.400 0.376 0.354 0.334 0.315 0.298 0.282 0.267 O. 182 0.171 0.160 O. I50 0.141 0.133 0.126 0.1 19 0.113 0.107 0.463 0.436 0.41 I 0.387 0.363 0.342 0.322 0.304 0.287’ 0.27 1

”Fiom 9.1.3 and Ref (10).

Computed from Ref (9).

length and width directions of the bar. To avoid end effects, the width dimension,

U:

of the bar should be equal t9 or greater than lOD, and the length at least 5 W between grips,

to allow for uniform tensile stress in the gaged area. A

thickness, T, of 4Do or greater is suggested. The calibration bar should be subjected to tension forces, Fd. The forces

should produce tensile stress in the gage ud = - and care

should be taken to ensure that no bending is present. The calibration stress, 04, should be smaller than 1/3u,,id. Apply

the calibration forces two times, once before drilling the hole, and once after, recording each time the strain gage output,

(t,)befolr and (t,),ficr. Obtain differences:

(4d = (s)nflcr - (&erom

The calibrated values of constants 3 and

B

are:

Fd WT

A i 5

2

= ( 6 3 ) c d i- ik 2 U d (c3)cnl - (Ci)cd 2UC-d B =

9.1.4 The stresses u, and uv computed above shall be

assigned to directions po and p

+

90". Since both

2

and

B

are negative, a tensile (+) residual stress wiii produce a compres- sive (-) relieved strain. The direction of the largest algebraic stress will closely coincide with the direction of the largest measured strain of opposite sign.

10.1.1.1

Material,

10. I.

1.2

Pertinent mechanical properties,

10.1.2

Location of strain gages,

1

O. 1.3 Model and type of strain gages used, 10.1.3.1 Strain gage geometry,

10.1.4 The method used to drill the hole, 10.1.5 Plot of strain versus depth for each gage,

10.1.6 Tabulation of strains cl, ct, and t 3 at all locations,

1

O. 1.7 Tabulation of stresses and direction of stresses at all

and locations. NOTE 5-If the calculated stresses U, or U,, or both, exceed one half

of the yield stress of the material, the stresses on the edge of the drilled

hole might exceed the eiastic limit of the material. Depending upon the

material, the inelastic behavior could affect the accuracy of the results.

10. Report

ll. Rwision and

1

1. 1

There is not enough information at this time to evaluate the precision and accuracy of the method. A round- robin test ppgram is being carried out by the Residual Stress

Measurements Committee of the Society for Experimental Stress Analysis. The precision of this test method will be

assessed when the results of the round robin b e o e available.

10.1

The report shall include the following:

10.1.1

Description of the test specimen,

REFERENCES

Rendler, N. J., and Vigness, I., "Hole-Drilling Strain Gage Method

of Measuring Residual Stresses," Experimental Mechanics, Vol 6,

No. 12, 1966.

Perry, C. C., and Lissner, H. R., Strain Gage Primer, McGraw-Hili

b o k Co., Inc., New York, NY, 1955

Kelsey, R. A., "Measuring Non-Uniform Residual Stresses by the

Hole Drilling Method," Proceedings, §?BA, Vol 14, No. I, 1956,

Sandifer, J. P., and Bowie, G. E., "Residual Stress by Blind Hole

Method with Off-Center Hole," Experimental Mechanics, Vol 18,

"Measurements of Residual Stresses by the Blind Hole Drilling

Method," Tech Note TN503, Measurements Group, Wendeli, NC.

Bush, A. J., and Kromer, E J., "Simplification of the Hole-iMiing

Method of Residual Stress Measurements," ZSA Transaction, Vol

pp. 181-194.

NO. 5, 1978, pp. 173-179.

112, NO. 3, 1973, pp. 249-260.

(7) Eeaney, E. M., "Accurate Measurements of Residuai Stress on Any

Steel Using the Centre Hole Method," STRAIN, Jounial BSSM,

(8) Flaman, M. T., "Investigation of üitra-H@ Speed Lhiiiing for

Residual Stress Measurements b y the Center Hole Method," Ex-

perimenial Mechanics, Vol 22, No. 1,

1

982, pp. 26-u).

(9) Schajer, 6. S., "Application of finite Element calculations to

Residual Stress Measurements," Journal of Engineering Materials

and Technology, Transactions, ASME, Vol 103, April 1981, pp.

(10) Redner, S., and Perry, C. C., "Factors Affecting the Accuracy of

Measurements Using the Blind Hole Drilling

of 7th International Conference Expenmen-

Vol 12, NO. 3, 1976, pp. 99-106.

157-163.

tal Stress Analysis, August 1982, pp. 604-616.

The Amerkm Society br Tasting and Matwi& tekris no position respecting the validity of any petent rhhts asseited in contmctlm

with m y item mentioned in this standard. Users of this standard are expessty adviseú that determination d the validity of any such patemt rigMS, and the risk of i n f r i m of such rights, are entirely their own respmsibiiity.

This standard is subject to revision at any time by the responsik technical conunittee and must be reviewed every f/ve years and

it rot revised, e i t k reapproved or withdrawn. Your comments are invited either fglpovision of this standard or for &ditional standards

grid should ka addressed to ASTM Headquarters. Your comments will receive careful consideration at a meeting of the respmsibie

technical m m & e e , which you may attend. if you feel that your comments have not received a fair W i n g you should &e your

views known to the ASTM Committee on Standards, 7976 Race St., Philadelphia, PA 19103.

A2 1

APPENDIX 2: Distribution of stresses in isotroDic materials after

drilling a hole

This appendix is a summary of "Flügge, Handbook of engineering

mechanics"

.

From fig.1 can be derived:

dX

Because the deformed line elements must fit together, the strain must be

compatible. It is found by eliminating u and v from (1):

From the plane stress situation follows:

The conditions of equilibrium are in case of zero body forces (fig.2):

a+,, au = o = o -+' au, if+,,

- + -

_{az ay} al: ay ( 4 ) Y I ! I D X C X fig.2

Equations ( 4 ) are I6er;tieally satisfied, wheri the s%resses are expressed

as derivatives of Airy's stress function @:

a2.p ax ay T z y = -

-

a2.p ax2 c y = - ₍₅₎

A2 2

From (2) in ( 3 ) and ( 4 ) can be derived:

+

which can be written as:

= o

From substituting (5) in ( 6 ) can be derived:

VPV* = 0

With polar coördinates follows:

Stresses can be determined with:

( 7 )

Stress fields havinp - polar s m e t r y

The analysis is particularly simple when the stress distribution is symmetrical about the origin of the coördinates. Then all stresses,

strains, displacements, and the stressfunction Q depend on r only, and

all partial derivatives with respect to 6’ are zero. Then:

has the general solution:

<p = A In r

+

Br? In r

+

Crz

+

D

where A , B, C , D are-arbitrary constants.The corresponding stresses

A2 3

Stress fields without polar smmetry

When stresses and strains are proportionally to cos2û, equation (10) has

the general solution:

with the stresses:

where n=2.

Circular hole in an infinite plate with uniaxial tension.

The stress function describing the disturbance of the uniform tension is

given by eqs.(l2) and eqs.(l4). In order to maintain uniform tension at

infinity all particular solutions must be excluded, which would yield

finite or infinite stresses for r,. This requires that B=C=O in eqs.(l2)

and A=B=O in eqs.(l4). The other constants follow from the boundary constants, and lead to the stress function:

= - 2 j 2 in r - 5 4 P ( 4r-2

+

24') cos 20 ₍₁₅₎

By superposing the corresponding stresses and the uniform tension can be

obtained :

A3 1 APPENDIX 3 t Q

",ii

O u

1

I I! II o U L

f

I

A4

1

A P P E N D I X 4

CHARLES W. BERT

School of Aerostxice cind Mech(inicu1 Engineering

Utiicersity of Okluhoniu, Norniun, Okla.

AND

GARY L. THOMPSON

Fl u t ter and S t ructural D y no ni ics G roti p

Aero Coninlander Dicision

North ‘4 t i i ericu n Rock wel 1 C o rpo r(i tio n , Normo n , Okla .

A seniidestriictive method has been developed for determining

the principal residual stresses and directions i n rectan~ii1:irly

orthotropic materials. The reduction equations are based upon a

set of fcinctions that describe the siirface strain-relaxation field

about a hole drilled to a limited depth into the material. Three

constants contained in the strain functions have, to be determined

by calibration tests; they are related to three general constants and

the elastic inaterial constants to establish applicability to an

orthotropic material. Expressions for the planar residiial-stress

coinpoiients in the material-symmetry directions are then devel-

oped, ;ind from hlohr’s stress circle, the l)rincipal reïitfiial strvssrs m t l directioiis are determined.

INT UCTION

ESIIIUAL S T R E S S E S are quite often a nemesis for the engineer of tocì;iy. \Uien neglected i n design, residiial stresses ~‘itii cause a

1):n-t to fail I>elow the desipi Ioutf or, iiiider fatigiie loadings, prior to tiie iiscfiil d e s i ~ i i life of the 1);it-t. Hesitliial stresses ;ire soiiietiriies