Eval Eng11

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TESTING LABORATORY

Acoustical and Vibration Analysis

Residual Stress Measurement, Strain Gauge Testing

Thermodynamic Testing

Data Acquisition, Software Design & Development

Testing Facilities, Instrumentation

Mechanical and Electronic Design engineering

SINT Technology srl - Via Giusti 22950041 Calenzano (FI) - Italia

Tel. +39 055 8826302Fax +39 055 [email protected] no. 04185870484Companies Register no. FI017-55501Fully paid-up registered capital 39,000Laboratory authorized by the Italian Ministry of Innovation, Universities and Research (Law 46/82, art. 4)

REST NSYSTEM FOR MEASURING RESIDUAL STRESS

BY THE HOLE-DRILLING METHOD

BACK CALCULATION MANUAL

Calenzano, Florence, Italy

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SYSTEM FOR MEASURING RESIDUAL STRESS BY THE HOLE-DRILLING METHOD


EVAL-Eng11 - 2 -

TABLE OF CONTENTS

1. RESIDUAL STRESS MEASUREMENT BY THE HOLE DRILLING METHOD ............... 41.1. Introduction .................................................................................................. 41.2. List of symbols ............................................................................................. 7

2. ASTM E837-08: UNIFORM AND NON-UNIFORM STRESS .......................................... 82.1. Introduction .................................................................................................. 82.2. ASTM E837-08: uniform stress distribution .................................................. 92.2.1.Thin specimen .............................................................................................. 92.2.2.Thick specimen .......................................................................................... 102.2.3. Intermediate thickness specimen ............................................................... 112.2.4.ASTM E837-08: uniform stress - Extension ............................................... 122.3. ASTM E 837-08: non-uniform stress distribution ........................................ 132.3.1.Calculation method .................................................................................... 132.3.2.ASTM E837-08: non-uniform stress - Extension ........................................ 14

3. INTEGRAL METHOD .................................................................................................. 163.1. Calculation method .................................................................................... 163.2. Integral method - Extension ....................................................................... 18

4. KOCKELMANNS METHOD ........................................................................................ 195. DESCRIPTION OF THE PUSHBUTTONS .................................................................. 226. METHODS OF STRAIN INTERPOLATION ON THE CALCULATION DOMAIN .......... 24

6.1. Polynomial Interpolation ............................................................................. 246.1.1.Method of optimizing the interpolant polynomial degree ............................ 246.2. No interpolation (None Selection) .............................................................. 25

7. STEP DISTRIBUTION ................................................................................................. 267.1. Constant Step Distribution ......................................................................... 267.2. Increasing Step Distribution ....................................................................... 277.3. Optimized Step Distribution ........................................................................ 27

8. DESCRIPTION OF THE CALCULATION METHODS .................................................. 28

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8.1. ASTM E837-08 Method: Uniform Stress .................................................... 288.2. ASTM E837-08 Method: Non-Uniform Stress ............................................ 298.3. Integral Method .......................................................................................... 308.4. Kockelmanns Method ................................................................................ 32

9. REFERENCES ............................................................................................................ 34

Read this Operating and Maintenance Manual carefully before startingto use the equipment.

Always keep this manual with the equipment.

Should you have any doubts or problems, contact the SINT Technologytechnical support team ([email protected]).
mailto:[email protected]:[email protected]:[email protected]:[email protected]

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1. RESIDUAL STRESS MEASUREMENT BY THE HOLE

DRILLING METHOD

1.1. Introduct ion

Residual stresses can be present in any mechanical structure because of manycauses: they may be due to the technological process used to realize the component(plastic deformation or welding), or could be caused by localized yielding of thematerial, i.e. because of a sharp notch, or to a particular kind of surface treatmentlike shot peening or surface hardening.

The residual stresses play the same role in the strength of a structure that commonmechanical stresses do but, while the stress due to external loads can be calculatedwith a certain accuracy, the residual stresses are difficult to foresee, and therefore itis very important to have a reliable method able to measure them directly in thestructure with a minimum damage for it.

Thats what hole drilling method (HDM) comes for. Basically, the HDM consists indrilling a small hole in the component material at the centre of a strain gauge rosette.The residual stresses, because of the removed material, relax and the surfacestrains can be measured by the strain gauges. Finally, a suitable mathematicalmodel evaluates the relaxed stress from the deformation measurements.

Currently, four main applications of the HDM exist in the Eval software: the ASTME837-08, relating to constant through the thickness stress field, the ASTM E837-08,relating to non-uniform stress through the thickness, a method proposed by H.Kockelmannbased on strain ratio measurements and the Integral methodrelating tovariable through thickness stress (proposed by G. S. Schajer).

The main functions of the Eval software are the following:

Best fit of strain values measured versus hole depth

Calculation of residual stress

For each stage there is a standard procedure, which the software executes asdefault, unless otherwise specified. Normally the default procedures are those

recommended.

With this system, a large number of depth increments can be achieved with highaccuracy making it possible to determine a curve of relieved strains through the useof the test points.

Application of a calculation procedure for a better interpolation may serve to improvethe stability and quality of the end result. The minimum number of hole-drillingincrements is 8, as indicated in Standard ASTM E837-08 (for uniform stressdistribution in the depth). However, the best results are obtained with 40 or more. Itshould, however, be noted that the minimum increment depth is 0.010 mm forsignificant increments for measuring the strain values between one increment and

another.

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In the polynomial interpolation, the calculation of the best interpolation (best fit) ofthe data is conditioned by determination of the polynomial coefficients, which is doneby the least squares method. It is also possible to select the degree of polynomialinterpolation for each set of data recorded for the three strains measured as afunction of hole depth. The software determines by default the best possible degreeof the interpolating polynomial through using the optimization procedure.

The software disposes of the following procedures for calculating residual stresses:

- Standard ASTM E837-08: uniform stress

- Standard ASTM E837-08: non-uniform stress

- Integral method

- Kockelmann method

The procedures have different fields of application and the following notes should betaken into consideration in deciding which is the most appropriate method.

Uniform Stress Method [Standard ASTM E 837-08]

This is the method described in standard ASTM E 837-08, based on the assumptionthat stresses do not vary with distance from the surface of the specimen. For thisreason, the method does not consider spatial resolution. Nevertheless, whenmeasured residual stresses are in fact uniform, this is the method to choose,

because it is the least sensitive to the effects of test errors.

Non-Uniform Stress Method [Standard ASTM E 837-08]

This is the method described in standard ASTM E 837-08, based on the assumptionthat stresses vary with distance from the surface of the specimen. Therefore, thespatial resolution is higher than with the other methods.

The numerical coefficients established by ASTM E 837-08 are used for thecalculation. The maximum depth that the method can be used for is 0.5 times themean radius of the strain rosette used for the test.

The ASTM E 837-08 standard establishes 20 acquisition depht steps in the firstmillimeter, each one of 0.05 mm.

Integral Method

This method provides a separate residual stress analysis at every hole drilling depthincrement. The integral method should be chosen when residual stresses areexpected to vary significantly with depth; however, it also has the highest sensitivityto test errors. This problem quickly gets worse when you try to raise the spatialresolution increasing the depth increments.

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The numerical coefficients established by Schajer are used for the calculation. Themaximum depth that the method can be used for is 0.5 times the mean radius of thestrain rosette used for the test.

The systems software allows you to select the number and distribution of depthincrements, while in the Non-Uniform Stress Method these parameters are fixed bythe ASTM standard. The distribution may be constant or variable; the variabledistribution reduces sensitivity to error through an increment amplitude optimizationprocedure. The option selected by default in the system is the optimized procedure.

Kockelmann method

This method uses the numerical coefficients calculated by Kockelmann and allowsyou to reach a calculation depth equal to the hole diameter. It is a method which haslittle sensitivity to the effects of test errors but it is valid only in a very particular caseof rosette diameter and hole diameter (dm/d0= 3).

Recently, also coefficients for different values of D/D0 have been provided andincluded in the Eval software.

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1.2. List of sym bols

D0 hole diameter

D rosette mean diameter

Rm rosette mean radius

s specimen thickness

z hole depth

Z axial position in depth

h adimensional hole depth = z/Rm

H adimensional position in depth = Z/Rm

adimensional depth = Z/D0

1, 2, 3 deformations measured by strain gauges 1,2,3

max maximum principal stress

min minimum principal stress

angle of orientation of the main stress

11 normal stress in gauge 1 direction

33 normal stress in gauge 3 direction13 shear stress on a surface normal to gauge 1 direction

angle between gauge 1 and principal stress direction

P (33 + 11) / 2

Q (33 - 11) / 2

T 13

p (3+ 1) / 2

q (3- 1) / 2

t (3+ 1- 22) / 2

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2. ASTM E837-08: UNIFORM AND NON-UNIFORM STRESS

2.1. Introduct ion

ASTM E837-08 ([1]) deals with the evaluation of both constant and not constantresidual stress field through the thickness of the specimen.

Despite the uniform stress condition does not occur very frequently (and it can beverified only after drilling the hole), its historical background makes it worth todedicate an introduction to it. Moreover, whenever the condition required to identifythe stress field as constant are fulfilled, it can give a quite accurate result.

The non-uniform stress condition is very common and the calculation describes the

residual stress profile in the depth: the ASTM E837-08 standard provides a staticcalculation method where drilling and calculation parameters are provided.

The ASTM E837-08 standard takes into consideration three different kinds of rosette(fig. 2.1), indicated as kind A, B and C.

Fig. 2.1 - Strain gauge rosettes used in ASTM E837-08

Stress measurement specifications are given for small thickness specimens (s < 0.4D) and thick specimens (s > 1.2 D), while suggestions are given for mid thicknesscomponents, though measurements are less accurate in this case.

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2.2. ASTM E837-08: unifo rm stress distr ibut io n

2.2.1. Thin specim en

The complete hole must be drilled through the thickness in one step and the strains

1, 2, 3 must be measured. Following, the combination of strains below must becalculated:

2

31 p

2

13 q

2

2 213 t (2.1)

From the combination of strains (p, q and t), the combination of stresses must becalculated as shown below:

)1(2P

y

a

Epx

b

Eqy

2Q

x

b

Etxy T

(2.2)

where a and b are calibration constants defined in tab. 2-A of the standard,according to the rosette geometry used.

The maximum and minimum principal stress value are given by:

22, TQPMINMAX (2.3)

The angle of the maximum principal stress is given by the equation1:

Q

Tarctan

2

1

(2.4)

The value of the principal angle is defined by the table shown below, depending on

the minus/plus sign of T and Q.

Tab 2-AValue of the principal angle

1

Since the atan function is defined only for 90

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2.2.2. Thick s pecim en

The hole must be drilled in 8 steps of 0.05 D partial depth each and the

measurements 1, 2, 3must be recorded.

For each drilling step, the strain combination (2.1) must be evaluated.

In order to check that the stress field is actually constant through thickness, asuitable test is given:

1. identify the numerically larger set of strain combinations between p, q and t;

2. express p,q and t as a percentage of their values at 0.4 D;

3. plot these data versus h and these diagrams should fall within a 3%tolerance range with the diagram shown in fig. 2.2. Points that do not fall

within this tolerance are either measurement errors or due to non uniformthrough thickness stresses.

Fig. 2.2 - Uniform stress test

If the stress relieved is find out to be actually constant, the following stresscombinations must be evaluated:

21 a

paEP

2b

qbEQ

2b

tbET

(2.5)

The maximum and minimum principal stresses can be evaluated as:

22

minmax, TQP

(2.6)

The principal angle can be evaluated using the same equation given in (2.4).

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WARNING: According to the standard ASTM E837-08 forintermediate hole the results are approximate.

2.2.3. Intermediate thickn ess sp ecimen

Intermediate thickness specimens are not within the scope of ASTM E837-08

standard, anyway some indications can be given when 0.4D

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2.2.4. A STM E837-08: un iform stres s - Extens ion

Calibration constants for blind and through hole in thick and thin specimens havebeen evaluated by SINT Technology for all the types of strain gage rosettesavailable on the market. The calculated calibration costants are helpful to increasethe precision of the residual stress evaluation with the EVAL software.

Tab. 2-BCalibration constants (ASTM E837-08: uniform stress distribution)

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2.3. ASTM E 837-08: non-uniform stress distr ibu t ion

2.3.1. Calcu lat ion metho d

The hole must be drilled in 40 steps of 0.05 mm partial depth each and the

measurements 1, 2, 3 must be recorded. Following, for each drilling step j, thecombination of strains below must be calculated:

2

)( 13 jjp

2

)( 13 jjq

2

)2( 213 jjt

(2.8)

From the combination of strains (p, q and t), the following combination of stressesmust be calculated for each drilling step:

pE

Pa

1 qEQb

tETb

(2.9)

The matricial system provides results for the values Pk, Qk and Tk. These valuesdepends on the residual stresses as shown in the equations below:

2

])()[( kxky

kP

2

])()[( kxky

kQ

kxykT )( (2.10)

Principal stresses and principal angle can be calculated by the following equations:

22)(,)( kkkkMINkMAX TQP (2.11)

k

kk

Q

Tarctan

2

1

(2.12)

The value of the principal angle is defined by table 2-A, depending on the minus/plussign of T and Q.

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2.3.2. ASTM E837-08: non-uniform stress - Extensio n

Calibration constants for non-uniform stresses distribution have been evaluated bySINT Technology for all the types of strain gage rosettes available on the market.The calculated calibration costants are helpful to increase the precision of theresidual stress evaluation with the EVAL software.

Tab. 2-CCalibration constants (ASTM E837-08: non-uniform stress distribution)

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3. INTEGRAL METHOD

3.1. Calculat ion metho d

Integral method for residual stress analysis was proposed by G. S. Schajer in 1988([4], [5] and [6]) in order to overcome the limits of ASTM standard regarding theconstant stress field.

The systems software allows you to select number and distribut ion of the depthincrements, while in the Non-Uniform Stress Method these parameters are fixed bythe ASTM standard.

Using the equations (2.1) and (2.5), it can be shown that the evaluation of residualstresses can be obtained by solving three separate integral equations like:

ih

ii dHHPhHAE

hp0

)(),(1

)(

ih

ii dHHQhHBE

hq0

)(),(1

)(

ih

ii dHHThHBE

ht0

)(),(1

)(

(3.1)

where A and B are suitable influence functions for equibiaxial and shear stresswhose aim is to take into account the effect of the stress relieved at depth Hof an h-depth hole on the strain gauges measurements.

G. S. Schajer didnt give the equations of A and B functions directly but, instead,used a different approach, called Integral Method.

In this method, the contributions to the total measured strain relaxations of thestresses at all depths are considered simultaneously.

In order to simplify the problem of residual stress evaluation, Schajer proposed that

the stress field could be described by means of step-wise functions whose value isconstant through the partial hole depths, in such a way that the integral equation(3.1), could be easily evaluated, provided that the influence function integrals couldbe calculated for each drilling step.

If this could be done, the (3.1)2assumes a discrete form as:

i

j

jjii PapE

1

,1

i

j

jjii QbqE1

,

i

j

jjii TbtE1

,

(3.2)

2

In order to keep things simple, in the following it will be referred only the P component of stress, evenif the same considerations can be applied also to the shear stress components.

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with 1 j i n

where nis the number of partial hole depths achieved during drilling stage and jia , is

the strain relaxation due to a unit Pstress within increment jof a hole i-increments

deep (fig. 3.1) and its relation with ),( hHA is:

i

i

H

H

iji dHhHAa

1

,, (3.3)

Fig. 3.1Coefficients physical meaning

The (3.2) is a linear system with a coefficient-matrix lower triangular, that can besolved with forward substitution.

With the aid of FEM calculations, G.S. Schajer made a mesh of the function

H

i dHhHAhhA0

,, (3.4)

from which its easy to evaluate the jia , coefficients

ijijji hHAhHAa ,, 1, (3.5)

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With reference to strain gauge rosette MM 062-RE, the functions A and B aregiven for D0/D=0.3, 0.4 and 0.5, and h=0,0.05,..,0.50 (tab. 3-A). Different values forD0/D and hrequire interpolation of the given coefficients.

3.2. Integral method - Extension

Calibration constants for Integral Method application have been evaluated by SINTTechnology for all the types of strain gage rosettes available on the market. Thecalculated calibration costants are helpful to increase the precision of the residualstress evaluation with the EVAL software.

Tab. 3-A - Schajer coefficients for MM 062-RE rosette

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4. KOCKELMANNS METHOD

This method, proposed by H. Kockelmann in 1993 [3], is based on the strain ratiomeasured during the hole drilling.

Fig. 4.1Hole shape examples

Kockelmann proposed, in addition to the standard hole obtained by high speed mill,a new hole shape to be obtained by electrochemical erosion (fig. 4.1).

Fig. 4.2Method description

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With reference to fig. 4.2, the method foresees a preliminary stage (to be realizedjust once for every rosette kind) for an experimental/numerical evaluation ofcalibration functions Kxand Ky, defined as

(4.1)

(4.2)

Where x and y are, respectively, the deformation measured by the strain gaugeoriented in the load direction and a 90 in a uniaxial loaded specimen.As an example, Kockelmann supply the Kxand Kyfunctions for a HBM rosette, kind1-RY61-1,5/120S (fig. 4.3), and for dm/d0= 3.

After the calibration functions have been defined, the stress field can be calculatedin a general case by the application of the following relations:

(4.3)

(4.4)

(4.5)

a) Rosette HBM 1-RY61-1,5/120S

b) Calibration functions

Fig. 4.3Kx and Ky function for HBM 1-RY61-1,5/120S rosette

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Equations (4.3)-(4.5) can be used in order to define the principal stresses and theangle of orientation:

(4.6)

(7.7)

These values are obtained with the aid of the Mohrs circle.

As written above, Kockelmann supply the Kxand Ky functions for a particular HBMrosette with D = 5.1 mm and for a particular hole diameter D 0(D/D0= 3).

However, this method is valid for all the standard HBM rosettes in the RSM software,since the key data are the same.

Recently, also coefficients for different values of D/D0 have been provided andinserted in the Eval software.

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5. DESCRIPTION OF THE PUSHBUTTONS

The main program appears on the display in the form shown in Fig. 6.1. Thefollowing pushbuttons are used:

- Load Data: This button opens the dialog box from which the operator choosesthe data file to be analyzed.

- Mod./Exp.Data: Opens a dialog box from which the operator can modify thepreviously loaded data. The modified data will be saved on file if the operatorpresses SAVE to exit. If the operator presses OK the modified data will besaved in memory only.

Fig. 5.1Residual Stress Evaluation Panel

- Export Stress Calc.: This pushbutton allows the user to export all stressinformation displayed on the window. The format of the output file isspreadsheet (text with tab separator). Export of stress information is available

for all calculation methods.

- Show/Hide Vectors: Show/Hide the window displaying the two vectors max

(direction of maximum stress), min (direction of minimum stress) and theirangular position in respect to the strain gage.

- Print: This command starts printing the results of calculations of the selectedmethod.

- EXIT:This button interrupts execution of the program.

- Description:This field contains a brief description of the selected test retrievedfrom the data file selected by the operator, which has previously been generatedthrough the Residual Stress Measurement System program. The user can

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modify the content of this field and save the changes made using the procedureassociated with the Mod./Exp. Data button.

- Date: Date of the selected test; it is retrieved from the data file selected byoperator. The user can modify the content of this field and save the changesmade using the procedure associated with the Mod./Exp. Data button.

- Material: The material code is retrieved from the data file selected by theoperator. The user can modify the content of this field and save the changesmade using the procedure associated with the Mod./Exp. Data button.

- Treatment:Description of material treatment retrieved from the data file selectedby the operator. The user can modify the content of this field and save thechanges made using the procedure associated with the Mod./Exp. Data button.

- Strain Gage: Designation of the type of strain gage rosette utilized, retrievedfrom the data file selected by the operator.

- Type, A/B/C:Designation of the shape of strain gauge rosette utilized; retrievedfrom the data file selected by the operator. The designation (A / B / C) isassigned according to ASTM E 837-08 [1].

- interpolation / Poly order / Number of Steps / Step / Calc. depth: SeeChapter 6 and 7.

- Endmill Diam. [mm]:Shows the value of the hole diameter (retrieved from thedata file selected by the operator).

- E [N/mm

2

]:Modulus of elasticity of the material expressed in N/mm

2

(retrievedfrom the data file selected by the operator).

- Eccentricity [mm]:Value of eccentricity in millimetres.

- Beta []:Angular direction of the eccentricity.

- :Value of Poisson's ratio (retrieved from the data file selected by the operator).

- S.G. Radius [mm]:Value of the mean radius of the strain gage rosette utilized inmillimetres.

- Hole Diam. [mm]: Value of the hole diameter (retrieved from the data fileselected by the operator).

- Show / Hide Legend:This button shows/hides plot legend.

- Hole: Blind/Through: This function is visible only when the ASTM E837-08method is selected. It allows hole type selection. The user can choose betweenBlindand Throughhole (see Chapter 2 for details).

- Calc: Quick/Precise: This function is visible only when the ASTM E837-08method is selected: the user can choose between Quick and Precise calculation.When Quick calculation is selected, a and b coefficients are calculatedevaluating strain distribution measured when the hole depth equals 0,4 times themean diameter of the strain gage. When Precise Calculation is selected, ASTM

a and b factors are coefficients from 0 to 0,4 Z/D.

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6. METHODS OF STRAIN INTERPOLATION ON THE

CALCULATION DOMAIN

In the Eval software, strains measured versus depth are interpolated on thecalculation domain in accordance with 2 distinct methodologies:

- Polynomial

- None

The fundamental principle followed in determining the strains on a different domainfrom the measurement domain is that of using calculation functions that identify thenearest interpolated points to the piecewise-linear obtained by joining themeasurement points. This principle pursues the objective of using a strain

distribution as near as possible to the measured distribution in calculating stresses.

Fig. 6.1 - Interpolation of measured strains

6.1. Polynom ial Interpolat ion

Polynomial interpolation is a regression on the quadratic minima with an nth degreepolynomial effected on the distribution of measured strain versus depth data.

The degree of the polynomial can be selected by the user and can be from 1 to 20,or it can be identified by the software with an optimum polynomial automatic searchfunction.

6.1.1. Method o f opt imizing the interpolant polyn om ial degree

To identify the optimum interpolant polynomial degree one proceeds by calculatingthe square deviation between the result of interpolation obtained with nth degreepolynomial regression and linear interpolation on the calculation domain.

The square deviation is calculated by variable polynomial degrees from 1 to 20 andthe polynomial degree that produces the smallest square deviation is taken asoptimal.

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Fig. 6.2Method of optimizing the interpolant polynomial degree

6.2. No interp olatio n (None Selection )

This method consists in evaluating strains on the measurement domain: by selectingthis method the calculation steps are set on Original by default and therefore no

regression or interpolation of the strains is applied.

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7. STEP DISTRIBUTION

The user can select the number of steps, the step distribution and the depth ofcalculation of residual stresses (expressed also as a percentage of the mean radiusof the strain gage rosette).

The depth of calculation is related to the mean radius Rm of the rosette used,excluding the ASTM method for which stresses are constant through the thickness.All calculation methods have a limit at a depth of 0.5 Rm, where the function ofinfluence becomes almost equal to zero. For this reason, the best results areobtained with a depth of 0.35 to 0.4 Rm, which corresponds to a depth of 0.9 to 1.1mm with standard rosettes (Rmequal to approx. 2.5 mm).

The number of calculation steps does not have an actual limitation, apart from theresolution of the instrument (approx. 10 micron); normally, however, good results areachieved with 10 to20steps.

Fig. 7.1Selection and distribution of calculation steps

7.1. Cons tant Step Distr ibu t ion

In the Constant mode, constant calculation steps throughout the depth of a holeused for calculation are applied. The set calculation depth is subdivided into intervals

corresponding to:

StepN

DepthCStep

_

_

(7.1)

where:

- C_Depth maximum calculation depth (normally derived as a percentage ofthe maximum measurement depth)

-

N_Step number of calculation steps

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Note: the point corresponding to null drilling is not considered in counting thecalculation steps.

7.2. Incr easing Step Distr ibu t ion

In the Increasing mode, a distribution corresponding to a cosinusoidal functiondeveloped on of period is used. The depth as a function of the angular domaincan be expressed as:

DepthCDepth ii _cos1 (7.2)

where:

- C_Depth maximum calculation depth (normally derived as apercentage of the maximum measurement depth)

- angle corresponding to the i-th calculation step

- N_Step number of calculation steps

- i index of variable steps between 1 and N_Step

In this way, a distribution of calculation steps is achieved with a greaterconcentration near the surface. The gradient of the function used progressivelyincreases as the depth increases.

7.3. Opt im ized Step Distr ibu t ion

This option, which was solely developed for calculation by the integral method inaccordance with the procedure specified in the article: Optimal selection of thenumber of steps for calculation of variable stresses in a thickness using the method

of the integral equation (D.Vangi, B.Zuccarello), is currently disenabled and should itbe selected Constant distribution is selected by default.

iStepN

i _

1

2

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8. DESCRIPTION OF THE CALCULATION METHODS

All the procedures for calculating stresses operate on a strain domain specified bythe operator which may be interpolated. One exception is the ASTM Method in whichthe calculation steps are set by the standard.

The interpolation can be viewed in detail by selecting the option Show Interpolationin the window for selecting the calculation methods (Fig. 8.1).

Fig. 8.1Interpolated strains

8.1. ASTM E837-08 Method: Unifo rm Stress

The ASTM E837-08: uniform stress can be used to analyze uniform stressconditions in isotropic elastic materials.

For further details on the procedure, refer to the standard [1] and Section 2.

The calculation procedure includes the first stage of finding the stress value withblind hole / intermediate hole / through hole and quick / precise calculation options(see Section 6, only for blind hole), and a second stage of verifying stressdistribution uniformity.

When Quick calculation is selected, a and b coefficients are calculated evaluatingstrain distribution measured when the hole depth equals 0.4 times the meandiameter of the strain gage. When Precise Calculation is selected, ASTM a and bfactors are coefficients from 0 to 0.4 z/D.

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Fig. 8.2Calculation by the ASTM Method: uniform stress

8.2. ASTM E837-08 Method: Non -Uniform Stress

The ASTM E837-08: non-uniform stress can be used to analyze a not uniformstress distribution in the depth of isotropic elastic materials.

For further details on the procedure, refer to the standard [1] and Section 2.

This calculation method is static because its fixed by the standard, and so its

impossible to set some evaluation parameters such as number of steps andcalculation depth.

The results show the residual stress profile in the depth: the left plot provides theprincipal stress distribution in the depth, and the right plot provides the equivalentstress by Von Mises and Tresca.

The button Show vector shows the direction of the principal angle in the depth.

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Fig. 8.3Calculation by the ASTM Method: non-uniform stress

8.3. Integral Method

The Integral calculation method has been developed following the indications ofSchajer ([4], [5] e [6]). This calculation is not set by the standard, and so in the rightwindow its possible to change some evaluation parameters such as as number ofsteps and calculation depth.

The following discussion is general and therefore applies also to other methods.

What changes is the way in which coefficients a and b are calculated. Thiscalculation with the integral method has already been described in detail in Section3.

The fundamental equations for strain used in the software are the following:

2

13 hhhp

2

13 hhhq

2

2 213 hhhht

where:

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mR

zh = adimensional hole depth (referred to the mean radius of the rosette R m).

Whereas the following are used for stress:

213 HHHP

2

13 HHHQ

HHT 13

where:

mR

ZH = adimensional depth from the surface (referred to the mean radius of the

rosette Rm).

By means of the above transformations, the bond between measured strain andstress, normally expressed in matrix form, can be split (and therefore considerablysimplified) into 3 separate equations, called bond equations:

hpE

HPa

1

hqEHQb htEHTb

where a and b are the matrixes of influence that contain the coefficientscorresponding to the relaxation function for a blind hole on material with uniform

stress.

Matrixes a and b are determined by Schajers matrixes, with interpolation on theadimensional depth domain used for the calculation. Schajers mat rixes weredetermined by a finite element calculation for a number of D0/D values as a functionof adimensional depth h (Section 3).

Solving the system of linear equations identified by the bond equations obtains thedistributions of the following variables:

P(H) Q(H) T(H)

which in turn make it possible to determine:

HQHPH 1 HQHPH 3 HTH 13

and especially the principal stresses:

22 HTHQHPHsx (maximum stress)

22 HTHQHPHsy (minimum stress)

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HQ

HTH arctan

2

1 (maximum stress angle, measured from gauge 1 to the

maximum principal stress direction. The positive direction is the one that takes thedirection of gauge 1 to the direction of gauge 3).

Fig. 8.4Calculation by the Integral Method

8.4. Kockelmanns Method

Kockelmanns method is based on the theory that there is a correlation functionbetween the strain derivative and the stress distribution, expressed as a function ofhole depth. The bond is formed by a pair of coefficients (K xand Ky), calculated on asimulation model, that relate stress and strain in accordance with the equations seenin Section 4.

From these stress values it is then possible to calculate the principal stresses andangle by using Mohrs Circle.

The Kxand Kyvalues are tabulated as a function of the adimensionalized depth ofhole z/D0and ratio D/D0. To simplify the calculation operations, the tables have been

approximated with polynomial functions expressed as a function of the

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adimensionalized depth. These polynomials are used in the calculation procedure tofind the values of Kxand Kywith the assigned D/D0, z and number of steps.

Fig. 8.5Calculation by Kockelmanns Method

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9. REFERENCES

[1] ASTM E 837-08, Standard Test Method for Determining Residual Stresses bythe Hole-Drilling Strain-Gage Method

[2] ASTM E 837-01, Standard Test Method for Determining Residual Stresses bythe Hole-Drilling Strain-Gage Method

[3] Schwarz, T., Kockelmann, H., The hole-drilling method - the best technique forthe experimental determination of residual stresses in many fields ofapplication, MTB 29, 1993, Vol. no. 2, pages 33-38

[4] Schajer, G. S., Measurement of Non-Uniform Residual Stresses Using theHole-Drilling Method. Part I - Stress Calculation Procedures, Journal of

Engineering Materials and Technology, Vol. no. 110, 1988, pages 338-343[5] Schajer, G. S., Measurement of Non-Uniform Residual Stresses Using the

Hole-Drilling Method. Part II - Practical Application of the Integral Method,Journal of Engineering Materials and Technology, Vol. no. 110, 1988, pages344-349

[6] Schajer, G. S., Application of Finite Element Calculations to Residual StressMeasurements, Journal of Engineering Materials and Technology, Vol. no.103, 1981, pages 157-163

[7] Soete, W., and Vancrombrugge, R., An Industrial Method for theDetermination of Residual Stresses, Proceedings SESA, Vol. 8, No. 1, 1950,

pages 17-28

[8] Kelsey, R. A., Measuring Non-Uniform Residual Stresses by the Hole DrillingMethod, Proceedings SESA, Vol. 14, No. 1, 1956, pages 181-194.

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