hole-drilling integral method with DOI: 10.1177 ...

Original article

J Strain Analysis2016, Vol. 51(6) 431–443� IMechE 2016

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First-order correction to counterthe effect of eccentricity on thehole-drilling integral method withstrain-gage rosettes

Michele Barsanti, Marco Beghini, Leonardo Bertini,Bernardo D Monelli and Ciro Santus

AbstractThe offset between the hole and the centre of the strain-gage rosette is unavoidable, although usually small, in the hole-drilling technique for residual stress evaluation. In this article, we revised the integral method described in the ASTME837 standard and we recalculated the calibration coefficients. The integral method was then extended by taking intoaccount the two eccentricity components, and a more general procedure was proposed including the first-order correc-tion. A numerical validation analysis was used to consolidate the procedure and evaluate the residual error after imple-menting the correction. The values of this error resulted limited to a few percentage points, even for eccentricitieslarger than the usual experimental values. The narrow eccentricity limit claimed by the standard, to keep the maximumerror lower than 10%, can now be considered extended by approximately a factor of 10, after implementing the pro-posed correcting procedure, proving that the effect of the eccentricity is mainly linear within a relatively large range.

KeywordsResidual stresses, hole-drilling method, integral method, eccentricity correction, finite element analysis

Date received: 8 January 2016; accepted: 12 April 2016

Introduction

Residual stresses can significantly influence the strengthof mechanical components, especially under cyclic fati-gue loading.1 There are several experimental techniquesfor measuring residual stresses.2 These include mechani-cal methods (destructive or semi-destructive) based oncuttings or local material removals to relieve theembedded residual stresses, such as the contourmethod,3 the ring core method4–6 and other specificapproaches adapted to the geometry of the componentsinvestigated,7–11 or also combining the hole-drillingmethod with the indentation.12 After the material hasbeen removed, a back calculation is required to obtainthe residual stress distribution that has beenrelaxed.13,14 The hole-drilling method is the most popu-lar and widely investigated.15,16 Its hardware is easilyimplemented, it is relatively inexpensive and it directlyprovides the stresses at a point; specifically, it averagesthe stress on the small volume of the removed material.Hole-drilling can be performed by measuring the entirefield of the relaxed deformation with optical methods17

both on the isotropic elastic materials, namely, steel or

other metallic alloys, and on the orthotropic materialssuch as composite plates.18 However, the hole-drillingtechnique is usually performed just with a strain-gagerosette to measure the relaxed strains in the near area ofthe drilled hole. There are many application examplesof this technique such as large components showing aflat (or almost flat) surface,19 deep rolled or shot peenedflat specimens,20,21 and recently even on coated sur-faces.22–25 The usual rosette for hole-drilling has threegrids angularly placed at 08, 2258, 908. Alternatively, the08, 458, 908 grid pattern can be used without making anydifference, for the angular periodicity of the stresses,unless a hole eccentricity is introduced. Figure 1 showsthe hole-drilling set-up with type A or type B three gridrosettes according to the American Society for Testing

Department of Civil and Industrial Engineering (DICI), University of Pisa,

Italy

Corresponding author:

Ciro Santus, Department of Civil and Industrial Engineering (DICI),

University of Pisa, Largo L. Lazzarino 2, 56122 Pisa, Italy.

Email: [email protected]

at Biblioteca di Scienze on August 28, 2016sdj.sagepub.comDownloaded from

http://sdj.sagepub.com/

and Materials (ASTM) standard. Besides the direction2 at 458, rather than at 2258, type B has a smaller gridwidth than grid length to prevent any geometryinterference.

After having measured and recorded the relaxedstrains, the integral method is usually considered for thecalculation of the residual stress distributions. The inte-gral method was initially proposed by Schajer26,27 andrecently has been widely used and also conformed tospecific cases, such as medium thick plates28 and thinplates,29 as well as being extended to optical methods.30

The ASTM E837-13a standard31 codifies the hole-drilling integral method, with a strain-gage rosette. Thecalculation proposed takes into account that a variabledistribution can be obtained, where the stresses areassumed to be piecewise constant over each hole depthincrement. Then, the standard reports the calibrationcoefficients (dimensionless and in matrix format) whichdepend only on the geometry, thus enabling the resi-dual stresses from the measured relaxed strains to becalculated.

Error analysis is crucial for any residual stressexperimental technique; indeed, we recently proposed abending test rig for providing a validation concurrentlywith the measure itself.32,33 Many investigations can becited from the literature considering different sourcesof errors for the hole-drilling technique. The strainmeasure uncertainty, which is the most important,34

can be reduced to some extent by following the regular-ization procedure proposed by Schajer35 and thenimplemented in the ASTM E837 starting from the 2008issue. Another source of error affecting the hole-drillingmeasures is the plasticity induced by the drilling,for which we proposed a correction in a previousstudy.36,37 The shape, radius and the position of thehole are also the reasons for experimental uncertainty.The flatness of the bottom surface was questioned byScafidi et al.38 and then by Nau and Scholtes,39 whilethe problem of eccentricity was initially investigatedanalytically by Ajovalasit40 for a thin workpiece where

a through-hole (plane stress) assumption can be used.More recently, Beghini et al.41,42 introduced the influ-ence function approach, for a blind hole in a thickworkpiece. The strain field was computed starting froma database of numerical solutions, implementing a spe-cific geometric configuration in which the componentsof eccentricity are merely introduced as the geometryparameters rather than being considered as a source oferror. The integral method according to the ASTME837 standard enables the residual stress calculation tobe easily implemented, but the presence of hole eccentri-city is a limitation. The standard requires a nearly per-fect concentricity between the hole and the rosette. Theprescribed maximum allowable eccentricity value is0:004D so that the maximum eccentricity is 0:02mmfor the usual rosettes with D=5:13mm. The currentpositioning accuracy is usually in the order of a fewhundredths of millimetres; thus, the eccentricity valuesare sometimes higher than this standard limitation. Thenumerical simulations, as reported below, showed thatan eccentricity in the order of these values can introducesome percentage points of error on the residual stresscomponents. In order to improve the integral method,this article proposes a correction procedure based onthe first-order linearization of the calibration matrices,which considerably extends the tolerable eccentricity.

Integral method

Within the concentricity assumption, the axial-symmetric geometry allows a decoupling between thecomponents of stress with respect to the correspondingrelaxed strains. The residual stresses are related to therelaxed strains according to the general relationshipintroduced by Schajer26,27

er(q)=A(smax+smin)+B(smax � smin) cos (2q)

ð1Þ

where smax and smin are the principal residual stresses,q is the angle of the generic rosette grid with respect tothe direction of smax and A and B are the two elasticconstants. After introducing a three-grid rosette with ageneric orientation with respect to the principal direc-tions, the following matrix equation holds

(A+B) (A� B) 0A A 2B

(A� B) (A+B) 0

24

35

sx

sy

txy

24

35=

e1e2e3

24

35 ð2Þ

in which 1, 2 and 3 are the three grid directions:08, 2258, 908 or 08, 458, 908. Given that directions 1 and 3are orthogonal and numbered as counter-clockwise(CCW), a reference system for the problem is inherentlyintroduced, and the axes x and y for the residual stresscomponents can be oriented according to the directions1 and 3, respectively (Figure 1). Alternatively, a clock-wise (CW) orientation could be introduced (in agree-ment with the ASTM E837 standard); however, theorientation of grid 3 would be opposite to the direction

Figure 1. Hole-drilling method with type A and type Brosettes according to the CCW numbering system.

432 Journal of Strain Analysis 51(6)



of the y-axis. The form of equation (2) suggests com-bining the variables according to an equibiaxial stress,and the related strain, plus two components of pureshear stress and the relative strains

P=(sy +sx)

2,Q=

(sy � sx)

2,T= txy

p=(e3 + e1)

2, q=

(e3 � e1)2

,

t=2e2 � (e3 + e1)

2= e2 � p

ð3Þ

In the second of equation (3), the component ofstrain t has been intentionally defined with the oppositesign with respect to the ASTM standard in order tohave sign consistency between the shear stress txy andthe relaxed strain e2. For example, when the equibiaxialstress component P is positive, then the strain p is nega-tive; indeed, the coefficients A and B are both negativesince the measured strains are the relaxed deformations.Similarly, when a positive shear stress txy is introduced,the principal stress along the 458 angle is tensile; there-fore, e2 turns out to be negative and thus t also has tobe negative. Having the grid direction 2 at 2258 makesno difference and the last of equation (3) is still valid.The opposite definition for the t strain componentwould be consistent for a CCW rosette with the grid 2direction at 1358, or alternatively at �458. However,this is never the case for the type of rosettes usuallyavailable, although the possible use of this scheme ismentioned by Nau and Scholtes.39 On the other hand,introducing the CW grid numbering system as shownin the ASTM standard, the second grid is either in thesecond quadrant (type A) or in the fourth quadrant(type B), and consequently, the opposite definition for tis correct, as reported in the standard. The numberingsystem CCW was considered preferential for this studysince it simplifies the sign definition of the eccentricitycomponents with both the two orthogonal grids 1 and3 oriented along the positive sign of the x- and y-axes.After substituting the definitions of equation (3) intoequation (2), the three variables are decoupled; thus,the matrix relating stresses and strains are diagonalcontaining the two coefficients A and B

2A 0 00 2B 00 0 2B

24

35

PQT

24

35=

pqt

2435 ð4Þ

This linear dependence between the residual stressesand the relaxed strains is a consequence of the elasticitybehaviour of the material. Therefore, equation (4) canbe rewritten with the explicit dependence of Young’smodulus E and also the negative sign of the two coeffi-cients can be emphasized

� 1

E

a 0 00 b 00 0 b

24

35

PQT

24

35=

pqt

2435 ð5Þ

where a and b depend on the Poisson’s ratio n and onthe ratios between the hole and the rosette dimensions,but they do not depend on Young’s modulus.

The stress–strain relation of equation (5) only con-siders a single value of the stress state, which is assumedto be uniform along the depth. In the specific case ofsmall thickness, plane stress, this relationship can befurther developed by making explicit the dependenceon the Poisson’s ratio n which only applies on the equi-biaxial stress component, thus obtaining equation (6)

� 1

E

a(1+ n) 0 00 b 00 0 b

24

35

PQT

24

35=

pqt

2435 ð6Þ

where, according to equation (5), a= a(1+ n),b= b.Now the coefficients a and b are scalar, positive, dimen-sionless and only depend on the geometry ratios, whilethey do not depend on the size. When the method isapplied to a surface of a bulk component (blind hole),and the state of stress can be assumed to be uniform upto the final depth of the hole, the dependence on thePoisson’s ratio according to equation (6) only has anapproximate validity. However, this effect can beneglected, since it produces a perturbation of the valuesin the order of a few percentage points, as discussedbelow. Thus, the dependency on n can also be kept forthe blind hole problem, as proposed in equation (6)which tolerates some inaccuracy; otherwise, a recalcula-tion of the coefficients is required for any value of n.

A major development of the method is the introduc-tion of a possible non-uniform distribution, in whichthe components of stress are introduced as uniformstepwise. There are thus three independent componentsfor each increment of the hole. In this case, the scalarsp, q, t and P,Q,T are replaced with vectors and thecoefficients a, b are replaced with the calibration coeffi-cient matrices

� 1+ n

E�aP= p, � 1

E�bQ= q, � 1

E�bT= t ð7Þ

Vectors P=(P(1),P(2), . . . ,P(n))T,Q,T and similarlyp=(p(1), p(2), . . . , p(n))T, q, t represent the stresses andstrains following the logic of decoupling, introducedpreviously, and i=1, 2, . . . n are the depth positionsalong the increments of the hole. The matrices �a and �bare lower triangular since the residual stress of thematerial removed in the previous steps has a (linear)cumulative effect on the relaxed strains, while the resi-dual stress of the material below, which still has to beremoved, has no effect on the relaxed strains.Therefore, the coefficients aij and bij of the matrices �aand �b are all positive for i5j, while they are 0 for i\ j.For example, just with three drilling steps, the coeffi-cient matrices are

�a=a11 0 0a21 a22 0a31 a32 a33

24

35, �b=

b11 0 0b21 b22 0b31 b32 b33

24

35 ð8Þ

Barsanti et al. 433



The integral method with variable stress distribu-tion, which is summarized in equation (7), just requiresthe axial symmetry of the removed material shape andthe orientation of the measurement grids according tothe 08, 2258(458), 908 scheme. Thus, the method can beextended to different problems that share these geo-metric characteristics, such as the ring core methodwhere material removal is annular and the grids aresuperimposed on the central volume according to thesame angular pattern.

Extension of the integral method to the eccentricity

When an eccentricity between the drilled hole and therosette is introduced, even if small, equation (1) is nolonger valid. Therefore, it is not possible to decouplethe p, q, t components and a new formulation isrequired. By following a more general approach, a lin-ear relationship between all the components of stressand strain can still be proposed, equation (9)

� 1

E�AS= e ð9Þ

where S=(s(1)x ,s(1)

y , t(1)xy , . . . ,s(n)x ,s(n)

y , t(n)xy )T is the vec-

tor of all the components of residual stresses, collectedin blocks of three terms, for different values of depth,and similarly e=(e(1)1 , e(1)2 , e(1)3 , . . . , e(n)1 , e(n)2 , e(n)3 )T is thevector of the relaxed strains. Since the three compo-nents of stresses and strains are arranged in blocks, �Ais a lower triangular 333 block matrix. An examplewith three depth increments is reported below

�A=

A(11)11 A

(11)12 A

(11)13 0 0 0 0 0 0

A(11)21 A

(11)22 A

(11)23 0 0 0 0 0 0

A(11) A(11) A(11) 0 0 0 0 0 0

A(21)11 A

(21)12 A

(21)13 A(22) A(22) A(22) 0 0 0

A(21)21 A

(21)22 A

(21)23 A(22) A(22) A(22) 0 0 0

A(21)31 A

(21)32 A

(21)33 A(22) A(22) A(22) 0 0 0

A(31)11 A

(31)12 A

(31)13 A(32) A(32) A(32) A(33) A(33) A(33)

A(31)21 A

(31)22 A

(31)23 A(32) A(32) A(32) A(33) A(33) A(33)

A(31)31 A

(31)32 A

(31)33 A(32) A(32) A(32) A(33) A(33) A(33)

2666666666666666666664

3777777777777777777775

ð10Þ

The coefficients A(ij)hk have the indexes: h, k=1, 2, 3

and i=1, . . . , n, j=1, . . . , i where, similarly to above,n is the total number of drilling steps. Again these coef-ficients do not depend on Young’s modulus, but dodependent on the Poisson’s ratio, and also on the geo-metry ratios. More specifically, each of these coeffi-cients also depends on the eccentricity. Therefore, thepower series expansion can be applied in terms of thetwo eccentricity components along directions 1 and 3

A(ij)hk =A

(ij)0, hk+

∂

∂e1A

(ij)hk e1 +

∂

∂e3A

(ij)hk e3 +

1

2

∂2

∂e21

A(ij)hk e

21 +

1

2

∂2

∂e23A

(ij)hk e

23 +

∂2

∂e1∂e3A

(ij)hk e1e3 + � � � ð11Þ

In equation (11), the two eccentricity components e1and e3 are introduced. Their definition intended here isthe offset of the hole with respect to the rosette, whichis positive according to the x,y-axes. Provided that x isrightwards and y upwards (Figure 1), when the holecentre is right-shifted, with respect to the grid centre, e1is positive; similarly, e3 is positive when the hole centreis up-shifted. The opposite definition of the eccentricitycomponents, or the use of the CW numbering system,would be equally possible but involving a revision ofthe derivative coefficient signs. As expected, and con-firmed by the numerical analysis reported below, thehigher order contributions are negligible, especially forrelatively small eccentricity values; thus, it suffices tofocus only on the first two linear terms. Furthermore,as matrix �A is dimensionless, the derivative terms canalso be put in a dimensionless form just by multiplyingby a characteristic length of the problem, for example,the average diameter D of the strain-gage rosette, equa-tion (12)

A(ij)hk =A

(ij)0, hk+a

(ij)1, hkh1 +a

(ij)3, hkh3 ð12Þ

where the dimensionless derivative coefficients and theeccentricity components are

a(ij)1, hk=D

∂

∂e1A

(ij)hk ,h1 =

e1D

a(ij)3, hk=D

∂

∂e3A

(ij)hk ,h3 =

e3D

ð13Þ

The coefficients A(ij)0, hk,a

(ij)1, hk,a

(ij)3, hk can be collected into

block triangular matrices

�A0 = ½A(ij)0, hk�, �ae1 = ½a(ij)

1, hk�, �ae3 = ½a(ij)3, hk� ð14Þ

and finally, matrix �A can be reconstructed, andapproximated to the first order, as

�A= �A0 + �ae1h1 + �ae3h3 ð15Þ

Clearly, the availability of matrix �A allows to solvethe inverse problem, that is, to determine the profile ofthe residual stresses from the measured relaxed strains,just by calculating the inverse matrix. Given the ratherlow size of the matrix, the inversion of �A does notinvolve numerical difficulties and the stress componentsS can be easily obtained from the measured strains e

S=�E�A�1e ð16Þ

Being the stress component decoupling not used, athree-grid rosette with 08, 2258(458), 908 pattern is nolonger strictly required. A different angular patterncould be proposed or even a higher number of gridscould be considered to increase the strain-gage sensitiv-ity. For example, with four grids instead of just three,the matrix would be 433 block and the inverse prob-lem solved with the pseudoinverse matrix rather thanthe square inverse. An example of a rosette with fourgrids is type D reported by Schajer,15 which has a




specific applicability when a correction of the plasticityeffect is required.36

Symmetry properties

As shown in Figure 2, the hole eccentricity affects thesensitivity of the grids. When the eccentricity is alongthe direction of the grid, the sensitivity is higher if thehole is closer to the grid; thus, the absolute value of themeasured relaxed strain is larger. On the other hand, ifthe eccentricity displacement is transversal, a portion ofthe grid has a higher sensitivity, while the other side hasa lower sensitivity and this implies, for symmetry, thatthe first-order derivative is 0. In fact, some of the coeffi-cients of the matrices �ae1, �ae3 turn out to be null. If anopposite grid is placed at the other side of the hole andconnected in series (Figure 2), another feature of sym-metry is introduced. Consequently, even the eccentricityfirst-order sensitivity along the grid direction is reducedto 0, and the derivative matrices are both entirely null.This situation is clearly of interest. In fact, the firstorder is systematically cancelled, and procedure p, q, tcan be followed without any notable effect caused bythe eccentricity. A commercially available rosette ofsix grids (two grids for each of the three directions)was tested by Beghini et al.43 and the reduced sensitiv-ity to the eccentricity was experimentally verified.However, introducing another grid for each measure-ment direction makes the rosette more complicated tomanage, and also more expensive. Thus, this kind ofrosette is not commonly used despite its inherentadvantage. Nau and Scholtes39 reported a rosette withup to eight grids and discussed how to connect themin different configurations and also highlighted thereduced sensitivity on the hole eccentricity. Therosette type C reported in the ASTM standard alsohas an opposite grid for each of the three directions;however, one side is radial and the other side is alongthe hoop direction. For this reason, the interactionis opposed to the compensation since the strain

measured on one side has the opposite sign withrespect to the other side; indeed, each couple ofstrain-gages has to be connected as half-bridge. Onthe contrary, the ring core method is another interest-ing example of compensating eccentricity. The threegrids are attached to the inside volume; thus, theeccentricity produces a smaller distance from theannular cut in one region and a higher distance inthe opposite region of the grid. Therefore, the sensi-tivity of each grid is compensated for and the first-order matrices �ae1, �ae3 are 0. This means that theprocedure p, q, t with decoupled components can be fol-lowed regardless of even relatively large eccentricities.

For common hole-drilling, without any compensat-ing extra grid, the problem reformulated according toequation (9) seems to require many terms. In reality,the symmetry properties reduce the number of freeparameters of the matrix �A and its derivatives, thusmaking the approach more accessible. Matrix �A0, whichis matrix �A in the case of no eccentricity, is a combina-tion of the coefficients of the two matrices �a and �b pre-viously introduced. It is thus an ‘inflated’ reissue (sinceit has more coefficients) of �a and �b. but offers a frame-work for the extension to the eccentricity problem. Thefollowing relationship (which is equivalent to equation(2)) offers the easiest way to calculate the coefficients ofthis matrix

A(ij)0, 11 A

(ij)0, 12 A

(ij)0, 13

A(ij)0, 21 A

(ij)0, 22 A

(ij)0, 23

A(ij)0, 31 A

(ij)0, 32 A

(ij)0, 33

2664

3775=

(1+ n)aij + bij2

(1+ n)aij�bij2 0

(1+ n)aij2

(1+ n)aij2 bij

(1+ n)aij�bij2

(1+ n)aij + bij2 0

2664

3775 ð17Þ

Equation (17) only shows a single block of the matrix�A0; anyway, this reconstruction is valid for all theblocks on the diagonal i= j and for any other blockbelow the diagonal i. j. A symmetry property alsorelates matrices �ae1 and �ae3. Given the angular pattern08, 2258(458), 908 of the grids, the derivation terms withrespect to the components of eccentricity are the samebut in different positions. Thus, one of the two can beobtained by the other just through simple permutationsof the coefficients. For example, if �ae1 is already avail-able, �ae3 is obtained by the following matrix equation(this relationship is only valid for a CCW system ofdirections 1 and 3)

�ae3 = �PL�ae1�PR ð18Þ

in which �PL and �PR are the two permutation left andright matrices, respectively, and they are 333 blockdiagonal containing only 0 and 1 in specific positions.The following is an example with three blocks of thesepermutation matrices

Figure 2. Grid sensitivity to longitudinal and transversaleccentricity components.

Barsanti et al. 435



�PL =

0 0 1 0 0 0 0 0 00 1 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 00 0 0 0 1 0 0 0 00 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 1 00 0 0 0 0 0 1 0 0

266666666666664

377777777777775

,

�PR =

0 1 0 0 0 0 0 0 01 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0

0 0 0 0 1 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 1 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 1

266666666666664

377777777777775

ð19Þ

Finite element modelling

A parametric finite element (FE) model was implemen-ted to determine the coefficients of the matrices intro-duced above. The axial-symmetric geometry enabledstructural plane harmonic elements (ANSYS Plane25)to be used. A full three-dimensional model, as pro-posed by Aoh and Wei44 and Xiao and Rong,45 was

avoided in order to produce a very high spatial resolu-tion in the region of interest with multiple nested refine-ments (Figure 3). The mesh division at the hole regionwas as small as one-hundredth of the radius, which wasassumed equal to 1mm as reference. The accuracy ofthe FE model was estimated to be approximately 1%,by comparing the numerical results with the analyticalsolution of the Kirsch equations, after reducing (justfor benchmarking) the model height to a single row ofelements and simulating a plane stress problem.

Two types of load were imposed on the axial-symmetric harmonic model: equibiaxial and pure shear.As reported in Figure 4, the equibiaxial was obtainedwith the harmonic order zero, while the pure shear wasobtained with the harmonic order 2, which is the ten-sorial dependence. To obtain the first load type, theremoval of material with residual stress embedded wassimulated as the application of pressure on the freecylindrical surface of the hole. Then, the pure shearwas obtained as pressure (symmetric with respect toq=0 angular coordinate) superimposed on a distribu-tion of tangential traction with a relative phase equalto 458 (anti-symmetric). These two load conditions,applied as uniform steps at different hole increments,enabled all the calibration coefficients to be obtained.Initially, without eccentricity, the procedure for aij andbij coefficients required the evaluation of p and q straincomponents, respectively, given by the combination ofdirections 1 and 3 alone. On the other hand, all thegrids were considered for the more general case witheccentricity, and the stresses sx and sy were obtainedas the subtraction and sum of the equibiaxial and shearcomponents, respectively.

According to equations (7) and (9), the calibrationcoefficients can be interpreted as the relaxed strainsinduced by unitary residual stresses. The strains mea-sured by the grids were numerically simulated from theFE analysis displacements by considering the angular q

dependence. The displacement fields on the upper

Figure 3. Plane FE model with axial-symmetric harmonicelements and multiple nested refinements.

Figure 4. Simulation of material removal for the equibiaxialand the pure shear stress load types.




surface are shown in Figure 5 for the two load types.The displacements along the grid direction, which isslightly different from the radial direction due to thegrid width, were calculated and averaged at severalintegration points along the two opposite sides of thegrid. The strain was obtained by computing the differ-ence of the averages at the two sides and finally bydividing by the grid length. This calculation is easierand faster and still equal to computing the averagestrain over the whole grid area, and for more detailssee, in particular, Nau and Scholtes.39 Finally, thisnumerical analysis was also performed with eccentri-city, following the same procedure, after updating theactual position of the grids with respect to the hole cen-tre. A square array of eccentricity component valueshas been considered in order to obtain the (numerical)derivatives of the calibration coefficients.

Calibration coefficient matrices

Standard calibration coefficients with no eccentricity

Although already available on the ASTM E837 stan-dard,31 the calibration matrices �a and �b were recalcu-lated as a benchmark of the numerical procedure, andalso to provide an update of the coefficients with thevery accurate FE model shown above. The exact com-bination of rosette dimensions, hole diameter and holedepth step is required to validate any single set of aijand bij calibration coefficients. The choice of otherdimension ratios involves different coefficients, whichneed the calculation to be repeated. In fact, the use ofan automated algorithm is recommended whereby anygeometry configuration can be inputted to generatethese coefficients.41,42 The specific dimensional combi-nations reported in the ASTM standard are consideredhere, and the calibration coefficients are provided inTables 1–4 of the online Appendix (available at: http://sdj.sagepub.com/), for type A and type B rosettes. TheASTM standard suggests the use of either millimetres

or inches as length units, however, with a simplifiedrounding. The coefficients according to both the stan-dard millimetres and inch dimensions were calculatedand can be found in the tables in the online Appendix.The effect of the approximated conversion was quanti-fied and the coefficient differences are in the order of afew percentage points up to 5%. Using inches, a com-parative analysis between the calculated coefficientswith this model and the values reported in the ASTMstandard is provided in Figure 6 for type A. The histo-grams highlight that the differences are limited to a fewpercentage points mainly at, and near to, the matrixdiagonal. Very similar comparison results were alsoobtained for type B.

The role of the Poisson’s ratio n as reported in equa-tion (7) is valid, in principle, only for the plane stressproblem, which is a very accurate approximation of thethrough-thickness hole on a thin plate. As discussedabove, the calibration coefficients actually have a moregeneral dependence with respect to the Poisson’s ratioand the plane stress form is only approximated. In orderto quantify the error introduced by assuming no depen-dence of the coefficients on the Poisson’s ratio, the FEanalysis was repeated with n=0:35 and the coefficientswere recalculated accordingly. This comparison analysisreturned similar coefficient values, and again two histo-grams with percentage differences are reported in Figure7. These differences are still in the order of a few per-centage points, and the highest discrepancies are for theinitial depths. Similar results were also obtained by Nauet al.46 confirming the validity of this analysis. The usualstructural metal alloys never show very large Poisson’sratio differences from the reference n=0:3 value. Thus,the form of equation (7) can be considered as satisfacto-rily accurate, after substituting the appropriate value ofn for the specific material, and using the calibrationcoefficients derived with n =0:3. In fact, all the coeffi-cients in the tables in the online Appendix are reportedfor reference n=0:3 alone.

Figure 5. Type A rosette, displacement fields for (a) the equibiaxial and (b) the pure shear load components.

Barsanti et al. 437



First-order correction for eccentricity

All the coefficients in matrix �A were calculated for anarray of eccentricities using directions 1 and 3 as refer-ence system. Initially, a large eccentricity range60:3mm was considered for the rosette diameterD=5:13mm. As an example, Figure 8(a) shows thedependence of a single coefficient of the matrix, andsimilar trends were obtained for all the other coeffi-cients. The dependency of the coefficients on the eccen-tricity is essentially linear thus, according to equation(11), the higher order terms introduce a further contri-bution which turns out to be minor, especially near thezero eccentricity origin. By reducing the investigatedeccentricity range to 60:05mm, which is still more thantwice the eccentricity allowed by the ASTM standard,Figure 8(b) highlights that the linear dependency is anextremely accurate model and the residual difference isnegligible. On the other hand, if a second compensatinggrid is introduced for each direction, or another self-compensating problem is investigated such as the ringcore method, the tangent plane at the origin is perfectly

horizontal as the first-order partial derivatives are 0. Inthese cases, the higher order terms still play a marginalrole, thus the p, q, t procedure is accurate and the pro-posed generalization is unnecessary. The matrix �ae1

was calculated by (numerically) determining the partialderivatives on the small range of Figure 8(b) and justby considering a single eccentricity component, specifi-cally direction 1. The coefficients of the matrix �ae1 arelisted in Tables 5 and 6 of the online Appendix (avail-able at: http://sdj.sagepub.com/), for the ASTM stan-dard type A and type B rosettes, respectively. Theother derivative matrix �ae3 can be easily calculatedfrom equation (18), and finally the matrix �A is obtainedwith equation (15). When a generic couple of eccentri-city components is introduced, the matrix �A has nomore 0 values at or below the block diagonal; hence,the proposed method is effectively exploited.

The Poisson’s ratio dependence of matrix �A0 is givenby equation (17). The appropriate value of n can beintroduced in the calculation of each block along withthe coefficients of the matrices �a and �b. However, the

Figure 7. Percentage differences for the Poisson’s ratio, aij and bij in (a) and (b), respectively, comparing n = 0:35 coefficients withthe reference n = 0:3.

Figure 6. Percentage differences obtained with respect to the ASTM type A, aij and bij in (a) and (b), respectively.




derivative matrix coefficient dependence on thePoisson’s ratio is more complicated since the equibiax-ial and the shear stress components are coupled. Thus,there is no simple equation to take into account thePoisson’s ratio for the derivative matrices, and the coef-ficients reported in Tables 5 and 6 of the onlineAppendix (available at: http://sdj.sagepub.com/), arejust defined for n=0:3. A recalculation would beneeded for any different values of n. However, thePoisson’s ratio is usually very close to n=0:3, at leastfor metals, as already discussed above. In addition, thederivative matrices only provide a correction for matrix�A; thus, the effect on the final result of a small varia-tion in the Poisson’s ratio is marginal.

Validation analysis

Numerical tests are proposed in this section to validatethe introduced eccentricity correction procedure. Usingthe algorithm developed by Beghini et al.41,42 based oninfluence functions, any residual stress distribution canbe simulated solving the direct problem and the relaxedstrains along the depth predicted. The residual stressdistribution, which is assumed as being uniform for thesake of simplicity, was initially back-calculated with theintegral method according to the standard, applying theusual p, q, t procedure and neglecting the eccentricity.The calculation was then repeated by introducing thelinear correction for matrix �A and solving the inverseproblem (equation (16)). Initially, a calculation examplewas performed with both the eccentricity componentsbeing not 0 and not equal. The eccentricities introducedwere in the order of a few hundredths of millimetres,similar to the values in real tests with D=5:13mmrosettes.32,47 The first calculation was implementedwith the type A rosette and eccentricity componentse1 =�0:02mm and e3 =0:05mm; thus, the overalleccentricity is larger than the limit prescribed by the

standard:ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie21 + e23

q. 0:02mm. The eccentricity intro-

duced a perceptible error with respect to the reference

stresses (Figure 9(a)); then this error was almost com-pletely eliminated after introducing the linear correction(Figure 9(b)), therefore confirming the effectiveness ofthe proposed procedure and the accuracy of the coeffi-cients provided. The second calculation was repeatedwith the same reference residual stresses but with type Brosette and just a single component of eccentricity,which, however, was 10 times larger than the limitallowed by the standard. Maximum stress error in theorder of 40% was obtained as the result of the p, q, tprocedure, thus meaning a considerable effect due tothe large eccentricity (Figure 10(a)). As highlighted inFigure 8, the linear component represents the actualtrend of the coefficients very well even for relativelylarge eccentricities. Thus, the stress reconstruction withthe linear correction was still quite accurate, returning amarginal residual error similar to the first examplewithout any correction (Figure 10(b)).

A wider analysis was then performed to test theeffect of eccentricity and verify the correction intro-duced by the procedure, covering all the combinationsof eccentricity values and stresses. Six levels of eccentri-city were considered: e1, e3 =0:02, 0:05, 0:07, 0:10, 0:15,0:20mm and with 16 angular positions for each:0:08, 22:58, . . . , 337:58. For all the combinations ofeccentricity radius and angle, three tests were separatelysimulated for the stresses sx,sy, txy=100MPa. Thehighest percentage difference was found along the depthcoordinate and saved for all the eccentricity and stresscombinations for both type A and type B rosettes. Thiscalculation was then repeated for the same combina-tions of eccentricity and stresses applying the correctionprocedure. Figure 11 highlights that the maximumeccentricity errors increased linearly with the eccentri-city, while the correction reduced the residual error bylimiting it to a few percentage points even for an eccen-tricity as large as 0:1� 0:15mm, which is quite highwith respect to the usual best practice in the experi-ments. The larger grid width of type A rosette averagesthe strain distribution better; thus, type B rosetteexperiences a higher eccentricity sensitivity when the

Figure 8. Eccentricity dependence of a single coefficient of matrix �A: (a) trend on a large range and (b) tangent plane on a small range.

Barsanti et al. 439



standard integral method is applied. With the proposedcorrection, the higher sensitivity of type B rosette isonly evident for large eccentricity values, while in therange where the linearity is predominant, the small resi-dual error is very similar for the two rosette types.

Conclusion

We have described a generalization of the hole-drilling integral method, which includes a correctionfor the eccentricity of the hole with respect to thestrain-gage rosette. The eccentricity impairs the axialsymmetry of the problem; thus, the decoupling of thestress components in an equibiaxial plus two shearstresses is no longer allowed. Consequently, a singlematrix is needed to linearly relate the relaxed strainsto the residual stress components. After having

grouped the strains and the stresses in vectors alongthe depth, this matrix is a lower triangular 3 3 3block, and each of its coefficients can be expressed asa power series of the eccentricity components. Thetwo linear terms alone already proved to be a veryaccurate model for reproducing the eccentricity effect.The matrix was then expressed as the zero eccentricityterm plus a linear correction of the two eccentricitycomponents by introducing the derivative matrices.In addition, the 08, 2258(458), 908 grid pattern onlyrequires the knowledge of a single derivative matrix,while the other can be obtained as a permutation ofthe coefficients, for symmetry reasons. On the otherhand, if a compensating grid is applied for each direc-tion, the first-order derivatives are 0 (similarly to thering core method) and the problem can still be formu-lated with the usual combined stresses and strains.

Figure 9. Validation example with type A rosette: (a) error caused by the eccentricity and (b) almost perfect result with the first-order correction.

Figure 10. Validation example with type B rosette: (a) error caused by a large eccentricity and (b) accurate result with the first-order correction.




A very refined plane and axial-symmetric harmo-nic FE model was implemented and the calibrationcoefficients were calculated. Initially, a revision of theASTM standard was proposed, according to the com-bined stresses and strains. The derivative matriceswere then calculated and provided for both type Aand type B strain-gage rosettes. Finally, an extensivenumerical analysis was proposed both to validate theprocedure and to show the accuracy of the correction.When the eccentricity is in the order of the smallallowed limit prescribed by the standard, the recon-struction of the stress components with the eccentri-city correction is very accurate. However, if theeccentricity is small, its measure uncertainty can be ofthe same entity of the eccentricity itself; thus, thepresent correction is not recommended. On the otherhand, quite accurate results were still obtained withlarger eccentricities, approximately up to 10 times thestandard limit, since the linear first order is a finemodel even for relatively high eccentricity values. Inconclusion, the proposed procedure can significantlyincrease the eccentricity allowed and, even when thehole is produced with a large offset for any experi-mental reason, the measure can still be useful insteadof being discarded or repeated.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research, authorship and/or publica-tion of this article.

Funding

The author(s) disclosed receipt of the following finan-cial support for the research, authorship, and/or publi-cation of this article: This work was supported by theUniversity of Pisa under the ‘PRA – Progetti di Ricercadi Ateneo’ (Institutional Research Grants) – Project No.PRA_2016_36.

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Appendix 1

Notation

�A calibration coefficient 333 block matrixa, b calibration coefficients relating the P,Q,T

residual stresses to the p, q, t relaxed strains�a, �b calibration coefficient matrices relating

the combined residual stresses and therelaxed strains

A, B general elastic constants relating theresidual stresses to the relaxed strains

D average diameter of the strain-gage rosettee vectors containing the blocks of the three

grid relaxed strains



http://sem-proceedings.com/09s/sem.org-SEM-2009-Ann-Conf-s044p03-Numerical-Study-Calibration-Coefficients-Hole-drilling-Residual.pdf





E, n Young’s modulus and Poisson’s ratio ofthe isotropic linear elastic material

h, k block indexes, for the matrix �A, rangingfrom 1 to 3

i, j calibration matrix depth indexes,i=1, . . . , n and j=1, . . . , i

n number of hole increments at the finaldepth

p, q, t combined relaxed strains according to theP,Q,T stresses

p, q, t vectors containing the combined strainsalong the depth arranged in blocks

P,Q,T equibiaxial and shear combined stressesP,Q,T vectors containing the combined stresses

along the depth arranged in blocksS vectors containing the blocks of the three

uncombined residual stress components

Greek symbols

�ae1, �ae3 derivative matrices for the calibrationcoefficient correction of 1 and 3directions, respectively

DZ hole depth incremental steper relaxed strain measured by a generic gride1, e2, e3 relaxed strains measured by the

08, 2258(458), 908 gridsh1,h3 dimensionless eccentricity componentssmax,smin principal maximum and minimum

residual stressessx,sy, txy residual stress components according to

the rosette reference frameq generic grid orientation with respect to

the maximum principal stress direction

Subscripts

�A0 calibration coefficient matrix �A with zeroeccentricity

D0 drilled hole diametere1, e3 eccentricity components along the

directions 1 (x) and 3 (y)GL,GW strain-gage rosette grid length and grid

width�PL, �PR left and right 333 block permutation

matrices

Barsanti et al. 443



Table 1. Type A rosette, ASTM E837–13a, D= 5.13 mm, GL = 1.59 mm, GW = 1.59 mm, D0 = 2.0 mm, ΔZ = 0.05 mm.Matrix a coefficients (×103)

6.613

7.932 7.099

9.040 8.359 7.296

10.018 9.360 8.540 7.291

10.886 10.230 9.473 8.521 7.132

11.654 10.994 10.263 9.400 8.342 6.856

12.331 11.666 10.945 10.125 9.172 8.036 6.491

12.926 12.256 11.538 10.740 9.840 8.818 7.633 6.065

13.447 12.771 12.054 11.267 10.397 9.434 8.367 7.159 5.598

13.900 13.219 12.502 11.721 10.868 9.940 8.935 7.846 6.640 5.110

14.294 13.609 12.889 12.113 11.270 10.363 9.395 8.369 7.279 6.093 4.614

14.635 13.946 13.224 12.449 11.613 10.719 9.774 8.786 7.758 6.686 5.538 4.124

14.929 14.237 13.513 12.738 11.907 11.021 10.091 9.127 8.137 7.125 6.085 4.986 3.649

15.183 14.488 13.762 12.987 12.157 11.278 10.358 9.410 8.443 7.467 6.485 5.489 4.449 3.194

15.402 14.704 13.976 13.200 12.372 11.496 10.583 9.645 8.695 7.742 6.795 5.853 4.909 3.933 2.766

15.590 14.890 14.160 13.383 12.555 11.681 10.773 9.843 8.903 7.967 7.041 6.132 5.239 4.353 3.445 2.366

15.752 15.049 14.318 13.539 12.711 11.839 10.934 10.008 9.077 8.151 7.241 6.352 5.490 4.651 3.826 2.988 1.996

15.891 15.186 14.453 13.673 12.845 11.973 11.070 10.148 9.222 8.304 7.404 6.530 5.687 4.876 4.095 3.333 2.563 1.656

16.010 15.304 14.569 13.788 12.959 12.088 11.186 10.266 9.344 8.431 7.538 6.674 5.845 5.052 4.297 3.574 2.874 2.172 1.347

16.112 15.405 14.668 13.886 13.056 12.185 11.284 10.366 9.446 8.537 7.649 6.792 5.972 5.192 4.453 3.754 3.091 2.452 1.814 1.067

Matrix b coefficients (×103)12.231

14.104 13.199

15.700 15.074 13.749

17.137 16.593 15.665 14.003

18.438 17.938 17.139 15.949 14.021

19.612 19.143 18.412 17.382 15.977 13.850

20.665 20.222 19.535 18.591 17.366 15.792 13.525

21.605 21.184 20.528 19.639 18.512 17.130 15.433 13.081

22.439 22.038 21.407 20.556 19.490 18.213 16.714 14.937 12.546

23.175 22.793 22.181 21.357 20.335 19.124 17.734 16.156 14.336 11.948

23.822 23.456 22.860 22.059 21.067 19.902 18.580 17.111 15.489 13.660 11.308

24.387 24.036 23.455 22.671 21.702 20.570 19.295 17.895 16.381 14.745 12.935 10.645

24.880 24.542 23.974 23.203 22.253 21.145 19.904 18.551 17.105 15.575 13.951 12.182 9.975

25.308 24.982 24.425 23.666 22.730 21.641 20.424 19.106 17.706 16.242 14.720 13.129 11.419 9.310

25.678 25.364 24.815 24.067 23.143 22.068 20.871 19.577 18.210 16.791 15.333 13.839 12.298 10.659 8.658

25.998 25.693 25.154 24.413 23.499 22.436 21.253 19.978 18.637 17.250 15.835 14.402 12.953 11.473 9.914 8.026

26.274 25.978 25.445 24.713 23.806 22.753 21.582 20.321 18.998 17.635 16.251 14.859 13.468 12.074 10.665 9.191 7.420

26.511 26.223 25.697 24.970 24.070 23.025 21.863 20.614 19.305 17.960 16.599 15.237 13.884 12.545 11.216 9.882 8.497 6.843

26.715 26.433 25.913 25.192 24.298 23.259 22.105 20.865 19.567 18.235 16.891 15.550 14.226 12.923 11.645 10.387 9.131 7.835 6.296

26.890 26.614 26.099 25.382 24.493 23.459 22.311 21.079 19.789 18.469 17.138 15.813 14.509 13.232 11.989 10.777 9.592 8.416 7.208 5.781

Table 2. Type A rosette, ASTM E837–13a, D= 0.202 in., GL = 0.062 in., GW = 0.062 in., D0 = 0.08 in., ΔZ = 0.002 in.Matrix a coefficients (×103)

6.967

8.366 7.477

9.542 8.813 7.676

10.580 9.874 8.995 7.658

11.500 10.796 9.983 8.961 7.475

12.314 11.605 10.817 9.890 8.754 7.167

13.030 12.315 11.538 10.654 9.630 8.412 6.766

13.659 12.936 12.163 11.302 10.332 9.235 7.967 6.301

14.207 13.479 12.705 11.856 10.917 9.882 8.739 7.450 5.795

14.683 13.950 13.175 12.332 11.412 10.411 9.333 8.169 6.885 5.268

15.096 14.358 13.581 12.741 11.832 10.853 9.813 8.714 7.553 6.296 4.737

15.453 14.710 13.931 13.093 12.189 11.225 10.208 9.148 8.051 6.913 5.699 4.215

15.760 15.014 14.232 13.394 12.494 11.539 10.537 9.502 8.444 7.367 6.267 5.110 3.711

16.025 15.275 14.490 13.652 12.754 11.804 10.813 9.795 8.761 7.722 6.680 5.630 4.539 3.231

16.252 15.499 14.712 13.872 12.976 12.030 11.046 10.038 9.021 8.005 6.999 6.004 5.013 3.993 2.781

16.447 15.691 14.902 14.061 13.165 12.221 11.241 10.241 9.235 8.236 7.253 6.291 5.352 4.424 3.479 2.362

16.614 15.856 15.065 14.222 13.326 12.383 11.407 10.412 9.413 8.425 7.457 6.517 5.609 4.730 3.870 3.000 1.977

16.757 15.997 15.204 14.360 13.463 12.521 11.546 10.555 9.562 8.581 7.624 6.699 5.810 4.960 4.144 3.352 2.557 1.625

16.879 16.117 15.322 14.477 13.580 12.638 11.664 10.675 9.686 8.710 7.761 6.846 5.971 5.139 4.349 3.598 2.874 2.151 1.306

16.984 16.220 15.424 14.578 13.680 12.737 11.764 10.777 9.790 8.818 7.874 6.966 6.100 5.281 4.508 3.781 3.093 2.435 1.781 1.019


14.858 13.887

16.554 15.880 14.456

18.082 17.493 16.492 14.706

19.463 18.920 18.055 16.772 14.704

20.707 20.197 19.404 18.290 16.776 14.497

21.821 21.338 20.591 19.569 18.245 16.552 14.128

22.814 22.354 21.640 20.674 19.454 17.964 16.143 13.633

23.692 23.253 22.565 21.639 20.483 19.104 17.491 15.588 13.044

24.465 24.046 23.377 22.481 21.370 20.060 18.561 16.868 14.926 12.392

25.143 24.740 24.089 23.215 22.136 20.874 19.447 17.868 16.133 14.187 11.699

25.733 25.347 24.711 23.854 22.800 21.571 20.193 18.686 17.064 15.319 13.400 10.985

26.246 25.874 25.251 24.409 23.373 22.170 20.827 19.369 17.818 16.183 14.457 12.587 10.267

26.691 26.331 25.719 24.890 23.869 22.685 21.368 19.945 18.442 16.876 15.256 13.571 11.768 9.558

27.074 26.726 26.124 25.305 24.296 23.128 21.830 20.433 18.964 17.445 15.891 14.306 12.679 10.957 8.866

27.405 27.067 26.473 25.663 24.664 23.508 22.225 20.848 19.404 17.918 16.409 14.887 13.355 11.798 10.165 8.199

27.689 27.360 26.774 25.971 24.981 23.834 22.563 21.201 19.776 18.315 16.837 15.358 13.885 12.418 10.939 9.401 7.561

27.933 27.611 27.032 26.236 25.252 24.113 22.852 21.502 20.092 18.648 17.194 15.746 14.313 12.901 11.506 10.111 8.669 6.956

28.142 27.827 27.254 26.463 25.485 24.353 23.100 21.758 20.359 18.930 17.494 16.067 14.663 13.288 11.945 10.628 9.319 7.974 6.385

28.321 28.012 27.443 26.657 25.684 24.558 23.311 21.977 20.587 19.169 17.746 16.335 14.952 13.604 12.296 11.027 9.790 8.568 7.318 5.848

Online Appendix:Michele Barsanti, Marco Beghini, Leonardo Bertini, Bernardo D Monelli, and Ciro Santus First-order correction to counter the effect of eccentricity on the hole-drilling integral method with strain-gage rosettesThe Journal of Strain Analysis for Engineering Design 0309324716649529, first published on June 8, 2016doi:10.1177/0309324716649529

Table 3. Type B rosette, ASTM E837–13a, D= 5.13 mm, GL = 1.59 mm, GW = 1.14 mm, D0 = 2.0 mm, ΔZ = 0.05 mm.Matrix a coefficients (×103)

7.017

8.423 7.531

9.605 8.874 7.734

10.649 9.941 9.059 7.718

11.574 10.868 10.053 9.029 7.538

12.392 11.681 10.893 9.964 8.825 7.233

13.112 12.395 11.618 10.733 9.706 8.486 6.834

13.744 13.021 12.247 11.385 10.414 9.315 8.044 6.370

14.296 13.567 12.793 11.943 11.003 9.966 8.821 7.528 5.865

14.776 14.041 13.266 12.423 11.501 10.500 9.420 8.253 6.964 5.339

15.192 14.453 13.675 12.835 11.925 10.946 9.904 8.803 7.638 6.375 4.808

15.551 14.808 14.028 13.190 12.286 11.321 10.303 9.241 8.141 6.998 5.779 4.285

15.861 15.114 14.332 13.494 12.594 11.638 10.636 9.599 8.538 7.458 6.353 5.189 3.780

16.128 15.378 14.593 13.754 12.857 11.906 10.915 9.895 8.859 7.816 6.771 5.715 4.617 3.299

16.357 15.604 14.817 13.977 13.081 12.134 11.150 10.141 9.122 8.103 7.094 6.095 5.098 4.071 2.848

16.554 15.798 15.009 14.168 13.272 12.328 11.348 10.347 9.339 8.337 7.351 6.386 5.441 4.508 3.556 2.427

16.723 15.965 15.173 14.331 13.435 12.492 11.515 10.519 9.519 8.529 7.558 6.615 5.702 4.818 3.952 3.075 2.040

16.867 16.107 15.314 14.471 13.574 12.632 11.657 10.665 9.670 8.687 7.728 6.799 5.907 5.052 4.231 3.433 2.630 1.687

16.992 16.229 15.435 14.590 13.693 12.751 11.777 10.787 9.796 8.819 7.867 6.949 6.070 5.234 4.440 3.683 2.952 2.221 1.365

17.098 16.334 15.538 14.692 13.794 12.852 11.879 10.890 9.902 8.929 7.982 7.071 6.202 5.378 4.601 3.869 3.176 2.510 1.848 1.076


15.532 14.529

17.313 16.622 15.144

18.920 18.318 17.285 15.427

20.374 19.822 18.931 17.602 15.445

21.687 21.169 20.354 19.203 17.631 15.250

22.864 22.374 21.608 20.554 19.182 17.420 14.883

23.914 23.449 22.718 21.724 20.461 18.913 17.014 14.384

24.845 24.402 23.698 22.745 21.552 20.121 18.443 16.456 13.786

25.665 25.243 24.561 23.639 22.493 21.136 19.579 17.814 15.782 13.118

26.386 25.982 25.318 24.420 23.308 22.002 20.521 18.878 17.066 15.027 12.407

27.015 26.628 25.980 25.102 24.015 22.745 21.316 19.750 18.058 16.234 14.219 11.672

27.563 27.191 26.557 25.694 24.628 23.385 21.993 20.479 18.863 17.156 15.348 13.383 10.931

28.038 27.680 27.058 26.208 25.158 23.936 22.572 21.095 19.531 17.897 16.203 14.435 12.537 10.198

28.450 28.103 27.492 26.653 25.616 24.410 23.067 21.618 20.091 18.508 16.884 15.224 13.514 11.698 9.481

28.805 28.470 27.867 27.038 26.012 24.819 23.492 22.064 20.564 19.016 17.441 15.848 14.241 12.602 10.878 8.789

29.111 28.785 28.191 27.370 26.353 25.170 23.857 22.445 20.965 19.444 17.902 16.356 14.812 13.270 11.712 10.084 8.126

29.375 29.057 28.470 27.656 26.646 25.473 24.169 22.770 21.306 19.804 18.288 16.775 15.274 13.792 12.324 10.851 9.323 7.496

29.601 29.291 28.711 27.902 26.899 25.732 24.437 23.048 21.596 20.110 18.613 17.123 15.654 14.212 12.800 11.412 10.028 8.599 6.901

29.795 29.492 28.917 28.114 27.115 25.955 24.667 23.286 21.844 20.369 18.887 17.415 15.968 14.556 13.182 11.846 10.540 9.245 7.915 6.341

Table 4. Type B rosette, ASTM E837–13a, D= 0.202 in., GL = 0.062 in., GW = 0.045 in., D0 = 0.08 in., ΔZ = 0.002 in.Matrix a coefficients (×103)

7.373

8.861 7.910

10.111 9.330 8.114

11.215 10.459 9.515 8.085

12.193 11.437 10.564 9.468 7.879

13.057 12.296 11.449 10.453 9.236 7.540

13.817 13.048 12.213 11.262 10.162 8.859 7.103

14.482 13.706 12.874 11.947 10.905 9.728 8.373 6.600

15.062 14.279 13.446 12.532 11.522 10.410 9.186 7.812 6.054

15.565 14.776 13.942 13.034 12.043 10.968 9.812 8.568 7.202 5.490

16.000 15.205 14.369 13.464 12.484 11.432 10.315 9.141 7.903 6.569 4.923

16.375 15.576 14.736 13.833 12.859 11.821 10.730 9.596 8.426 7.215 5.931 4.368

16.697 15.894 15.052 14.148 13.179 12.150 11.075 9.966 8.836 7.691 6.525 5.304 3.834

16.974 16.167 15.322 14.418 13.450 12.428 11.363 10.271 9.167 8.060 6.956 5.846 4.698 3.329

17.211 16.401 15.554 14.648 13.682 12.663 11.605 10.525 9.437 8.356 7.288 6.236 5.192 4.123 2.856

17.414 16.601 15.751 14.845 13.878 12.862 11.809 10.736 9.660 8.595 7.551 6.534 5.544 4.571 3.582 2.418

17.588 16.773 15.920 15.012 14.046 13.030 11.980 10.913 9.845 8.792 7.764 6.769 5.811 4.887 3.987 3.080 2.017

17.737 16.919 16.065 15.155 14.188 13.173 12.125 11.061 9.999 8.953 7.936 6.957 6.019 5.125 4.271 3.445 2.617 1.651

17.864 17.044 16.188 15.277 14.309 13.294 12.247 11.186 10.127 9.087 8.078 7.108 6.185 5.310 4.483 3.698 2.945 2.194 1.321

17.973 17.151 16.293 15.381 14.412 13.397 12.351 11.291 10.235 9.199 8.194 7.232 6.319 5.456 4.647 3.887 3.171 2.487 1.810 1.024


16.285 15.213

18.169 17.426 15.847

19.867 19.218 18.109 16.123

21.403 20.805 19.846 18.420 16.117

22.786 22.225 21.346 20.107 18.421 15.883

24.025 23.493 22.665 21.527 20.052 18.167 15.468

25.128 24.621 23.829 22.754 21.394 19.734 17.707 14.915

26.103 25.620 24.855 23.824 22.536 20.998 19.202 17.086 14.261

26.960 26.498 25.756 24.758 23.519 22.058 20.388 18.504 16.347 13.536

27.710 27.268 26.545 25.571 24.368 22.959 21.369 19.612 17.684 15.526 12.770

28.364 27.939 27.233 26.278 25.102 23.731 22.194 20.517 18.715 16.780 14.654 11.983

28.932 28.523 27.830 26.892 25.736 24.393 22.895 21.272 19.548 17.735 15.824 13.757 11.195

29.423 29.028 28.348 27.423 26.284 24.962 23.493 21.909 20.237 18.500 16.706 14.844 12.855 10.417

29.847 29.464 28.796 27.882 26.756 25.451 24.003 22.447 20.814 19.129 17.408 15.657 13.862 11.965 9.661

30.212 29.841 29.181 28.278 27.162 25.871 24.440 22.905 21.300 19.651 17.980 16.298 14.608 12.894 11.098 8.934

30.526 30.164 29.513 28.618 27.512 26.231 24.813 23.295 21.711 20.089 18.453 16.818 15.194 13.578 11.952 10.262 8.240

30.795 30.442 29.799 28.910 27.812 26.540 25.132 23.627 22.059 20.457 18.847 17.246 15.666 14.111 12.578 11.047 9.465 7.583

31.026 30.680 30.043 29.161 28.069 26.804 25.405 23.911 22.355 20.769 19.177 17.601 16.052 14.539 13.063 11.618 10.183 8.709 6.964

31.224 30.885 30.253 29.376 28.289 27.031 25.639 24.152 22.606 21.032 19.456 17.897 16.372 14.888 13.451 12.059 10.704 9.366 7.997 6.384

0.07245 -0.00773 0

-0.00535 -0.04035 -0.05661

0 0 -0.04942

0.087771 -0.00793 0 0.077439 -0.00867 0

-0.00796 -0.0484 -0.06756 -0.00516 -0.04339 -0.06079

0 0 -0.0571 0 0 -0.05399

0.100818 -0.00815 0 0.09242 -0.00924 0 0.078126 -0.00947 0

-0.01014 -0.05528 -0.07692 -0.00749 -0.05124 -0.07177 -0.00407 -0.04441 -0.06185

0 0 -0.06374 0 0 -0.06177 0 0 -0.05698

0.112371 -0.0084 0 0.104266 -0.00975 0 0.092992 -0.0103 0 0.075669 -0.01013 0

-0.01198 -0.06141 -0.08525 -0.00922 -0.0575 -0.08049 -0.00617 -0.05222 -0.07292 -0.00231 -0.04397 -0.06059

0 0 -0.06979 0 0 -0.06817 0 0 -0.06503 0 0 -0.05884

0.122506 -0.00866 0 0.114437 -0.01024 0 0.103911 -0.01098 0 0.090196 -0.01112 0 0.070864 -0.01063 0

-0.0135 -0.06686 -0.09259 -0.01061 -0.06296 -0.08801 -0.00755 -0.05806 -0.08112 -0.00416 -0.05169 -0.07154 -0.00011 -0.04244 -0.05756

0 0 -0.07534 0 0 -0.07391 0 0 -0.07132 0 0 -0.06713 0 0 -0.05979

0.131265 -0.0089 0 0.123165 -0.01068 0 0.11292 -0.01159 0 0.100216 -0.0119 0 0.084766 -0.01171 0 0.064409 -0.01095 0

-0.01471 -0.07167 -0.09894 -0.01168 -0.06773 -0.09449 -0.00856 -0.06298 -0.08791 -0.00522 -0.05715 -0.07917 -0.00164 -0.04996 -0.06814 0.002346 -0.04012 -0.05323

0 0 -0.08042 0 0 -0.07913 0 0 -0.07684 0 0 -0.07334 0 0 -0.06824 0 0 -0.05999

0.138715 -0.00911 0 0.130571 -0.01107 0 0.120436 -0.01213 0 0.108173 -0.01257 0 0.093855 -0.01253 0 0.077429 -0.01206 0 0.056927 -0.01108 0

-0.01562 -0.07587 -0.10435 -0.01248 -0.07189 -0.09999 -0.00928 -0.0672 -0.09359 -0.00591 -0.06161 -0.08526 -0.0024 -0.05505 -0.07513 0.001163 -0.04735 -0.06322 0.004876 -0.03728 -0.04805

0 0 -0.08505 0 0 -0.08388 0 0 -0.08178 0 0 -0.07866 0 0 -0.07436 0 0 -0.06852 0 0 -0.05956

0.144953 -0.00928 0 0.136769 -0.01139 0 0.126672 -0.01258 0 0.11462 -0.01313 0 0.100822 -0.0132 0 0.085556 -0.01287 0 0.068873 -0.01219 0 0.048961 -0.01104 0

-0.01628 -0.0795 -0.10887 -0.01303 -0.07549 -0.10459 -0.00975 -0.07082 -0.09832 -0.00632 -0.06535 -0.09021 -0.0028 -0.05909 -0.08053 0.000699 -0.05205 -0.06953 0.004065 -0.04413 -0.05727 0.007331 -0.03414 -0.0424

0 0 -0.08926 0 0 -0.08819 0 0 -0.08623 0 0 -0.08336 0 0 -0.07949 0 0 -0.07451 0 0 -0.06808 0 0 -0.0586

0.150101 -0.00939 0 0.141882 -0.01165 0 0.13179 -0.01294 0 0.119843 -0.01359 0 0.106311 -0.01375 0 0.091587 -0.01352 0 0.076029 -0.01296 0 0.05971 -0.0121 0 0.040952 -0.01084 0

-0.01671 -0.08262 -0.11259 -0.01337 -0.07859 -0.10839 -0.01001 -0.07391 -0.10219 -0.00652 -0.06851 -0.09423 -0.00297 -0.06242 -0.0848 0.000547 -0.05571 -0.07425 0.003877 -0.04845 -0.06287 0.006896 -0.04056 -0.05074 0.009597 -0.0309 -0.03661

0 0 -0.09305 0 0 -0.09209 0 0 -0.09024 0 0 -0.08754 0 0 -0.08396 0 0 -0.07945 0 0 -0.07392 0 0 -0.06704 0 0 -0.05722

0.154289 -0.00947 0 0.146042 -0.01184 0 0.135942 -0.01323 0 0.124044 -0.01396 0 0.110654 -0.01419 0 0.096214 -0.01403 0 0.081183 -0.01356 0 0.065912 -0.01281 0 0.050446 -0.01183 0 0.033238 -0.01051 0

-0.01696 -0.08528 -0.11561 -0.01353 -0.08123 -0.11146 -0.0101 -0.07655 -0.10533 -0.00656 -0.07119 -0.09746 -0.00296 -0.06518 -0.08818 0.000593 -0.05866 -0.07788 0.003944 -0.05174 -0.06694 0.006961 -0.0445 -0.05563 0.009526 -0.03684 -0.04401 0.011594 -0.02768 -0.03093

0 0 -0.09647 0 0 -0.09559 0 0 -0.09384 0 0 -0.09127 0 0 -0.08788 0 0 -0.08369 0 0 -0.07866 0 0 -0.0727 0 0 -0.06551 0 0 -0.0555

0.157651 -0.0095 0 0.149382 -0.01197 0 0.139267 -0.01345 0 0.12739 -0.01425 0 0.114075 -0.01454 0 0.099791 -0.01444 0 0.085039 -0.01402 0 0.070257 -0.01335 0 0.055738 -0.01248 0 0.041475 -0.0114 0 0.026053 -0.01006 0

-0.01705 -0.08754 -0.11801 -0.01355 -0.08347 -0.11392 -0.01006 -0.07879 -0.10783 -0.00646 -0.07345 -0.10002 -0.00281 -0.0675 -0.09084 0.000777 -0.06109 -0.08069 0.004167 -0.05436 -0.06999 0.007219 -0.04745 -0.05909 0.009817 -0.04041 -0.04821 0.011864 -0.03314 -0.03738 0.013276 -0.0246 -0.02553

0 0 -0.09953 0 0 -0.09873 0 0 -0.09706 0 0 -0.09459 0 0 -0.09136 0 0 -0.08738 0 0 -0.08267 0 0 -0.07723 0 0 -0.07097 0 0 -0.0636 0 0 -0.05354

0.160314 -0.0095 0 0.152027 -0.01205 0 0.141896 -0.0136 0 0.130023 -0.01446 0 0.116746 -0.0148 0 0.102549 -0.01475 0 0.087952 -0.01438 0 0.073428 -0.01377 0 0.059346 -0.01295 0 0.045916 -0.01198 0 0.033074 -0.01086 0 0.019542 -0.00952 0

-0.01703 -0.08945 -0.11989 -0.01346 -0.08537 -0.11585 -0.00992 -0.08068 -0.1098 -0.00627 -0.07535 -0.10203 -0.00258 -0.06944 -0.09292 0.001055 -0.06309 -0.08286 0.004487 -0.05649 -0.0723 0.007586 -0.04977 -0.06162 0.010237 -0.04305 -0.0511 0.012352 -0.03636 -0.04093 0.013853 -0.02958 -0.03106 0.014625 -0.02173 -0.02055

0 0 -0.10226 0 0 -0.10153 0 0 -0.09993 0 0 -0.09755 0 0 -0.09443 0 0 -0.09061 0 0 -0.08613 0 0 -0.08103 0 0 -0.07529 0 0 -0.06884 0 0 -0.06138 0 0 -0.05139

0.162397 -0.00947 0 0.154094 -0.01209 0 0.143947 -0.0137 0 0.13207 -0.01461 0 0.118811 -0.01499 0 0.10466 -0.01498 0 0.090149 -0.01465 0 0.075768 -0.01407 0 0.061915 -0.01331 0 0.048864 -0.01239 0 0.036736 -0.01136 0 0.025419 -0.01022 0 0.013778 -0.00892 0

-0.01692 -0.09106 -0.12134 -0.0133 -0.08697 -0.11734 -0.0097 -0.08228 -0.11132 -0.00601 -0.07695 -0.10358 -0.00228 -0.07106 -0.09451 0.001395 -0.06476 -0.08452 0.00487 -0.05823 -0.07405 0.008014 -0.05163 -0.06349 0.01072 -0.0451 -0.05317 0.012904 -0.03871 -0.0433 0.014508 -0.03247 -0.03401 0.015471 -0.02623 -0.0252 0.015646 -0.0191 -0.01605

0 0 -0.10468 0 0 -0.10402 0 0 -0.10249 0 0 -0.10018 0 0 -0.09715 0 0 -0.09345 0 0 -0.08914 0 0 -0.08427 0 0 -0.07887 0 0 -0.07293 0 0 -0.06639 0 0 -0.05895 0 0 -0.04912

0.164004 -0.00941 0 0.155687 -0.0121 0 0.145525 -0.01375 0 0.133639 -0.0147 0 0.120384 -0.01512 0 0.106257 -0.01514 0 0.091793 -0.01485 0 0.077492 -0.0143 0 0.063764 -0.01357 0 0.050909 -0.01269 0 0.0391 -0.01171 0 0.028381 -0.01065 0 0.018603 -0.00952 0 0.008777 -0.00827 0

-0.01674 -0.09241 -0.12243 -0.01308 -0.08831 -0.11847 -0.00944 -0.08362 -0.11247 -0.00571 -0.0783 -0.10476 -0.00194 -0.07242 -0.09571 0.001773 -0.06616 -0.08576 0.005286 -0.05967 -0.07535 0.008474 -0.05315 -0.06487 0.01123 -0.04673 -0.05466 0.013478 -0.04052 -0.04496 0.015167 -0.03455 -0.03593 0.016263 -0.02882 -0.0276 0.01672 -0.02316 -0.01989 0.016358 -0.01673 -0.01206

0 0 -0.10683 0 0 -0.10624 0 0 -0.10476 0 0 -0.10251 0 0 -0.09955 0 0 -0.09595 0 0 -0.09177 0 0 -0.08706 0 0 -0.0819 0 0 -0.0763 0 0 -0.07027 0 0 -0.06372 0 0 -0.05637 0 0 -0.04678

0.165226 -0.00934 0 0.156896 -0.01207 0 0.146719 -0.01377 0 0.134823 -0.01475 0 0.121565 -0.01521 0 0.107447 -0.01525 0 0.093007 -0.01498 0 0.078749 -0.01446 0 0.065091 -0.01375 0 0.05234 -0.01291 0 0.040692 -0.01196 0 0.030234 -0.01095 0 0.020944 -0.00988 0 0.012654 -0.00878 0 0.00452 -0.0076 0

-0.01653 -0.09354 -0.12325 -0.01282 -0.08944 -0.1193 -0.00914 -0.08475 -0.11333 -0.00538 -0.07943 -0.10563 -0.00157 -0.07357 -0.09661 0.002169 -0.06732 -0.08668 0.005718 -0.06087 -0.0763 0.008944 -0.0544 -0.06587 0.011745 -0.04805 -0.05573 0.014049 -0.04195 -0.04612 0.01581 -0.03615 -0.03723 0.017006 -0.03066 -0.02912 0.017621 -0.02546 -0.0218 0.017619 -0.02037 -0.01516 0.016791 -0.01463 -0.00857

0 0 -0.10874 0 0 -0.1082 0 0 -0.10677 0 0 -0.10457 0 0 -0.10168 0 0 -0.09815 0 0 -0.09407 0 0 -0.08949 0 0 -0.0845 0 0 -0.07913 0 0 -0.07343 0 0 -0.06738 0 0 -0.0609 0 0 -0.05371 0 0 -0.04442

0.166139 -0.00926 0 0.157798 -0.01203 0 0.147607 -0.01376 0 0.135699 -0.01477 0 0.122435 -0.01525 0 0.108317 -0.01532 0 0.093888 -0.01506 0 0.079651 -0.01456 0 0.066029 -0.01388 0 0.053335 -0.01305 0 0.041771 -0.01213 0 0.03144 -0.01115 0 0.022356 -0.01013 0 0.014452 -0.00908 0 0.007553 -0.00802 0 0.000957 -0.00692 0

-0.01628 -0.09448 -0.12383 -0.01254 -0.09039 -0.11991 -0.00883 -0.0857 -0.11395 -0.00504 -0.08038 -0.10627 -0.0012 -0.07452 -0.09725 0.00257 -0.06829 -0.08734 0.006149 -0.06186 -0.07699 0.009411 -0.05543 -0.06658 0.012252 -0.04914 -0.05648 0.014603 -0.0431 -0.04693 0.016424 -0.0374 -0.03811 0.017698 -0.03207 -0.03011 0.018426 -0.02709 -0.02297 0.018604 -0.02242 -0.01664 0.0182 -0.01788 -0.01102 0.016979 -0.01277 -0.00557

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0 0 -0.14252 0 0 -0.14187 0 0 -0.13998 0 0 -0.13703 0 0 -0.13315 0 0 -0.12847 0 0 -0.12312 0 0 -0.11722 0 0 -0.11091 0 0 -0.1043 0 0 -0.09749 0 0 -0.09055 0 0 -0.08352 0 0 -0.07642 0 0 -0.06922 0 0 -0.0618 0 0 -0.05384 0 0 -0.04386

0.191514 -0.01676 0 0.182047 -0.01984 0 0.170392 -0.02165 0 0.156733 -0.02258 0 0.141514 -0.02283 0 0.125341 -0.02257 0 0.108854 -0.02192 0 0.092649 -0.02097 0 0.077209 -0.0198 0 0.062895 -0.01849 0 0.049936 -0.01709 0 0.038444 -0.01565 0 0.028444 -0.01421 0 0.019889 -0.01278 0 0.012686 -0.0114 0 0.006716 -0.01006 0 0.001836 -0.00877 0 -0.00213 -0.00751 0 -0.00558 -0.00623 0

0.010696 0.112664 0.146983 0.006488 0.108025 0.142489 0.002405 0.102618 0.135555 -0.0017 0.096442 0.126584 -0.00579 0.089621 0.116041 -0.00975 0.082361 0.104458 -0.01345 0.074892 0.092371 -0.01676 0.067438 0.080267 -0.01959 0.060186 0.068548 -0.02187 0.053282 0.057517 -0.02358 0.046823 0.047377 -0.02472 0.04086 0.038242 -0.02532 0.035412 0.030153 -0.02542 0.030465 0.023099 -0.02505 0.025979 0.017027 -0.02424 0.021896 0.011859 -0.02301 0.018123 0.007497 -0.0213 0.014494 0.003809 -0.01874 0.010386 0.000457

0 0 -0.1439 0 0 -0.1433 0 0 -0.14145 0 0 -0.13854 0 0 -0.1347 0 0 -0.13006 0 0 -0.12476 0 0 -0.11892 0 0 -0.11269 0 0 -0.10618 0 0 -0.09948 0 0 -0.09268 0 0 -0.08584 0 0 -0.07899 0 0 -0.07214 0 0 -0.06523 0 0 -0.05816 0 0 -0.05061 0 0 -0.04118

0.191783 -0.0167 0 0.182308 -0.01981 0 0.170643 -0.02163 0 0.156974 -0.02257 0 0.141746 -0.02284 0 0.125564 -0.02259 0 0.109072 -0.02195 0 0.092861 -0.02101 0 0.077419 -0.01985 0 0.063104 -0.01855 0 0.050146 -0.01717 0 0.038659 -0.01574 0 0.028667 -0.01431 0 0.020125 -0.0129 0 0.012944 -0.01153 0 0.007007 -0.01022 0 0.002185 -0.00897 0 -0.00166 -0.00778 0 -0.00467 -0.00663 0 -0.00715 -0.00548 0

0.010383 0.113207 0.147133 0.006151 0.108573 0.142646 0.002047 0.10317 0.135716 -0.00208 0.096996 0.126748 -0.00619 0.09018 0.116209 -0.01017 0.082925 0.104629 -0.01389 0.075466 0.092545 -0.01722 0.068024 0.080445 -0.02008 0.060789 0.068731 -0.02239 0.053907 0.057708 -0.02413 0.047476 0.047577 -0.02532 0.041551 0.038453 -0.02597 0.036151 0.030381 -0.02613 0.031266 0.023349 -0.02585 0.026865 0.017306 -0.02516 0.022899 0.012181 -0.02409 0.019305 0.007887 -0.02266 0.015992 0.004327 -0.02079 0.012809 0.001387 -0.01812 0.009195 -0.00118

0 0 -0.14511 0 0 -0.14455 0 0 -0.14273 0 0 -0.13986 0 0 -0.13605 0 0 -0.13145 0 0 -0.12619 0 0 -0.12041 0 0 -0.11424 0 0 -0.10779 0 0 -0.10119 0 0 -0.0945 0 0 -0.0878 0 0 -0.08114 0 0 -0.07452 0 0 -0.06794 0 0 -0.06135 0 0 -0.05464 0 0 -0.04752 0 0 -0.03863

Table 5. Type A rosette, ASTM E837–13a, D= 5.13mm, GL = 1.59mm, GW = 1.59mm, D0 = 2.0mm, ΔZ = 0.05mm, coefficients of the derivative matrix αe1.

Table 6. Type B rosette, ASTM E837–13a, D= 5.13mm, GL = 1.59mm, GW = 1.14mm, D0 = 2.0mm, ΔZ = 0.05mm, coefficients of the derivative matrix αe1.

hole-drilling integral method with DOI: 10.1177 ...

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