**6. Eccentricity error: description and possible corrections**

The eccentricity between the drilled hole and the strain gage circle greatly influences the strain measurements. The ASTM standard requires a near perfect concentricity between the drilled hole and the rosette and prescribes an allowable eccentricity value that depends on the dimension of the strain gage rosette (0.004D). Using a standard rosette with a gage circle diameter D = 5.13 mm, the maximum allowable eccentricity is 0.02 mm. This limit increases (0.04 mm) or decreases (0.01 mm) using bigger (approx. D = 10.26 mm) or smaller types of rosette (approx. D = 2.56 mm).

As shown in **Figure 4**, the eccentricity error is influenced by the eccentricity coordinates ex and ey and by the parameters of the strain gage rosette (D, GL and GW). The type B rosette generally shows a higher sensitivity to eccentricity errors compared to type A rosettes; this can be explained by the orientation of the gage grids, which are concentrated only in the first quadrant (0°, 45°, 90°), instead of in the first and third quadrant (0°, 90°, 225°).

For these reasons, the correction of eccentricity errors requires accurate determination of the position of the drilled hole in the reference system of the strain gage rosette; eccentricity can be measured by a special procedure using the drilling system microscope. Using a digital microscope, the eccentricity coordinates can be easily obtained by image analysis techniques.

As shown in **Figure 5**, the sensitivity of the grids is directly influenced by the hole eccentricity. When the eccentricity has the same direction as the grid, if the hole is closer to the grid the absolute value of the relaxed strain is greater. On the contrary, if the eccentricity has a transverse direction with respect to the grid, then a portion of the grid has a greater sensitivity, while the other portion has a lower sensitivity; this implies, by symmetry, that the error is almost compensated [13].

The eccentricity correction can be done using strain gage rosettes with special configurations or using correction algorithms.

The studies of Beghini et al. [10] and of Nau et al. [11] introduced the correction of eccentricity using a special six-grid rosette and an eight-grid rosette respectively. Both the rosettes are produced by HBM and make it possible to compensate the first-order of eccentricity error. However, the corrections based on special strain gage geometries do not correct the higher order eccentricity errors (higher than

**Figure 4.** *Eccentricity in hole-drilling measurements.*

A ¼

**drilling method**

plasticity effect.

thick workpiece are considered.

consider possible corrections.

**Typical source of**

Intermediate thickness(s)

**Table 2.**

**66**

*methodologies.*

**error**

Að Þ <sup>11</sup> <sup>11</sup> <sup>A</sup>ð Þ <sup>11</sup>

Að Þ <sup>11</sup> <sup>21</sup> <sup>A</sup>ð Þ <sup>11</sup>

Að Þ <sup>11</sup> <sup>31</sup> <sup>A</sup>ð Þ <sup>11</sup>

Að Þ k1 <sup>11</sup> <sup>A</sup>ð Þ k1

Að Þ k1 <sup>21</sup> <sup>A</sup>ð Þ k1

Að Þ k1 <sup>31</sup> <sup>A</sup>ð Þ k1

<sup>12</sup> <sup>A</sup>ð Þ <sup>11</sup>

*New Challenges in Residual Stress Measurements and Evaluation*

<sup>22</sup> <sup>A</sup>ð Þ <sup>11</sup>

<sup>32</sup> <sup>A</sup>ð Þ <sup>11</sup>

<sup>12</sup> <sup>A</sup>ð Þ k1

<sup>22</sup> <sup>A</sup>ð Þ k1

<sup>32</sup> <sup>A</sup>ð Þ k1

ASTM standard and that of the new generalized calibration matrix.

<sup>13</sup> 000

<sup>23</sup> ⋯ ⋯ 000

<sup>33</sup> 000

<sup>11</sup> <sup>A</sup>ð Þ kk

<sup>21</sup> <sup>A</sup>ð Þ kk

<sup>31</sup> <sup>A</sup>ð Þ kk

<sup>12</sup> <sup>A</sup>ð Þ kk 13

(7)

<sup>22</sup> <sup>A</sup>ð Þ kk 23

<sup>32</sup> <sup>A</sup>ð Þ kk 33

⋮ ⋱ *:* ⋮ ⋮ *:* ⋱ ⋮

<sup>13</sup> <sup>A</sup>ð Þ kk

<sup>23</sup> ⋯ ⋯ <sup>A</sup>ð Þ kk

<sup>33</sup> <sup>A</sup>ð Þ kk

**Figure 3** compares the visual interpretation of the calibration matrices of the

The hole-drilling method has some typical sources of error that can influence the accuracy of the measurements. The ASTM E837 standard identifies the maximum values of these errors for the validity of the test (limits of applicability) without providing recommendations on how to correct them, as reported in **Table 2**.

Some of these errors, for example, eccentricity and hole bottom chamfer, can be generated by external sources such as the operator, testing condition or the drilling process. In other cases, the limits of applicability are directly connected with the test method, as in the case of intermediate thickness of the specimen or the local

**Table 2** also shows the bibliographic sources dealing with possible methods of error correction. The cases of uniform and non-uniform stress distribution are analyzed. In the case of uniform stress distribution, both the thin workpiece and the

The following sections examine the errors mentioned above in more detail and

**Suggested correction**

[12] [10, 11] [8, 9]

Thickness s ≤ 0.2D or s ≥ 1.0D [8, 9, 15, 16] [8, 9, 16]

applicable

**Thin Thick /**

/ [20] /

[10, 11]

**Uniform Non-uniform**

[8, 9, 17] [8, 9]

[8, 9, 13, 14] [10, 11]

**ASTM E837 limits of applicability**

Hole eccentricity Eccentricity radius within 0.004D

Local plasticity Magnitude of the stresses

Hole-bottom chamfer Not provided Not

≤50% of σY—Thin specimen ≤80% of σY—Thick specimen

*Typical sources of errors, ASTM E837 limit of applicability and current state-of-the-art of correction*

**5. The main sources of errors and limits of applicability in the hole**

(s ≤ 0.2D) the plane stress solution holds, the in-depth residual stress gradient is neglected and the through-hole method is applied; the influence coefficients for the thin plates can be directly deduced by Kirsch's solution of a membrane with a

*Recent Advancements in the Hole-Drilling Strain-Gage Method for Determining Residual Stresses*

For the intermediate thickness case (0.2D < s ≤ 1.0D), out-of-plane bending occurs (**Figure 6**) and this affects the calibration coefficients *a* and *b* defined in the ASTM standard. The calibration coefficients will depend directly on the workpiece

A preliminary solution to this effect for the case of a uniform stress field was proposed by Abraham and Schajer [15]. They provide an analytical model of the calibration coefficients a, b for intermediate thickness, as a function of workpiece

Recently, Beghini et al. [16] described a procedure for the evaluation of nonuniform residual stress for the intermediate thickness range. The authors define two equations (one for the coefficients aj,k and one for the coefficientsbj,k*),* that adequately reproduce the thickness dependency of all the ASTM E837 calibration

As reported in **Figure 7**, the dependency of the thickness is greater in the first part of the intermediate area (from 0.1D to 0.5D) and is less if the thickness

Moreover, a recent development of the generalized integral method based on the Influence Functions [8, 9] has introduced a new database of numerical solutions that takes thickness into account as an input parameter. The numerical database is

coefficients, for calculation of non-uniform stresses (**Figure 7**).

*Localized bending caused by hole-drilling in an "intermediate" thickness specimen.*

*Calibration coefficients as a function of plate thickness: for aij (left) and for bij, (right) coefficients in the*

circular hole.

is higher.

**Figure 6.**

**Figure 7.**

**69**

*matrices (from Beghini et al. [16]).*

thickness s, which becomes the new parameter.

thickness, hole diameter and hole depth.

*DOI: http://dx.doi.org/10.5772/intechopen.90392*

**Figure 5.**

*Grid sensitivity to longitudinal and transversal eccentricity and first-order eccentricity error compensation using a 6-grid rosette.*

0.2 mm), for which rosettes are required with bigger dimensions, a higher number of grids and higher costs.

Regarding the correction algorithms, the first solution was provided analytically by Ajovalasit et al. [12] for uniform stress in thin workpieces.

Beghini et al. [8, 9] provided a complete solution for blind holes using a generalized integral method based on the influence functions for non-uniform calculations (Section 4). According to this approach, the strain field is computed starting from a database of numerical solutions in which the eccentricity is simply introduced as a geometry parameter; this has the advantage of taking into account the whole effect of eccentricity. Recently, Barsanti et al. [13] proposed a simplified approach for the analytical correction of the first-order eccentricity errors in calculated stresses.

Peral et al. [14] has also proposed a correction approach applied to acquire strains.

### **7. Intermediate thickness limitation: description and possible correction**

The ASTM E837 method defines the application ranges concerning the thickness of the workpiece under testing. The measurements can be carried out on "thin" or "thick" workpieces, the thickness of which depends on the size of rosette. For a "thin" workpiece, the thickness should be less than 0.20D (for type A and type B rosettes) and the stresses are evaluated according to the "uniform stress calculation". For a "thick" workpiece the thickness should be greater than 1.0D (for type A and type B rosettes) and the standard provides the calculation methods for uniform and non-uniform stress distributions.

The range of thicknesses between 0.2D and 1.0D, defined as intermediate thickness, is outside the scope of the ASTM standard. Using a strain gage rosette with a gage circle diameter D = 5.13 mm, the intermediate thickness is identified in the range between 1 and 5.13 mm. Clearly, on varying the diameter of the strain gage circle, also the range of the intermediate thickness varies.

Unfortunately, intermediate thickness is common in several types of engineering applications, as in aerospace, motor sports and energy production.

This limitation in the ASTM standard can be explained by analysis of the behavior of stress response if a hole is made in an intermediate thickness specimen.

In the case of thick workpieces (s > 1.0D) the influence coefficients are independent of the thickness and they can be obtained by an FE model in which the hole is produced in a virtually semi-infinite body. In the case of thin workpieces

*Recent Advancements in the Hole-Drilling Strain-Gage Method for Determining Residual Stresses DOI: http://dx.doi.org/10.5772/intechopen.90392*

(s ≤ 0.2D) the plane stress solution holds, the in-depth residual stress gradient is neglected and the through-hole method is applied; the influence coefficients for the thin plates can be directly deduced by Kirsch's solution of a membrane with a circular hole.

For the intermediate thickness case (0.2D < s ≤ 1.0D), out-of-plane bending occurs (**Figure 6**) and this affects the calibration coefficients *a* and *b* defined in the ASTM standard. The calibration coefficients will depend directly on the workpiece thickness s, which becomes the new parameter.

A preliminary solution to this effect for the case of a uniform stress field was proposed by Abraham and Schajer [15]. They provide an analytical model of the calibration coefficients a, b for intermediate thickness, as a function of workpiece thickness, hole diameter and hole depth.

Recently, Beghini et al. [16] described a procedure for the evaluation of nonuniform residual stress for the intermediate thickness range. The authors define two equations (one for the coefficients aj,k and one for the coefficientsbj,k*),* that adequately reproduce the thickness dependency of all the ASTM E837 calibration coefficients, for calculation of non-uniform stresses (**Figure 7**).

As reported in **Figure 7**, the dependency of the thickness is greater in the first part of the intermediate area (from 0.1D to 0.5D) and is less if the thickness is higher.

Moreover, a recent development of the generalized integral method based on the Influence Functions [8, 9] has introduced a new database of numerical solutions that takes thickness into account as an input parameter. The numerical database is

**Figure 6.**

0.2 mm), for which rosettes are required with bigger dimensions, a higher number

*Grid sensitivity to longitudinal and transversal eccentricity and first-order eccentricity error compensation using*

by Ajovalasit et al. [12] for uniform stress in thin workpieces.

*New Challenges in Residual Stress Measurements and Evaluation*

Regarding the correction algorithms, the first solution was provided analytically

Beghini et al. [8, 9] provided a complete solution for blind holes using a generalized integral method based on the influence functions for non-uniform calculations (Section 4). According to this approach, the strain field is computed starting from a database of numerical solutions in which the eccentricity is simply introduced as a geometry parameter; this has the advantage of taking into account the whole effect of eccentricity. Recently, Barsanti et al. [13] proposed a simplified approach for the analytical correction of the first-order eccentricity errors in

Peral et al. [14] has also proposed a correction approach applied to

**7. Intermediate thickness limitation: description and possible correction**

The ASTM E837 method defines the application ranges concerning the thickness of the workpiece under testing. The measurements can be carried out on "thin" or "thick" workpieces, the thickness of which depends on the size of rosette. For a "thin" workpiece, the thickness should be less than 0.20D (for type A and type B rosettes) and the stresses are evaluated according to the "uniform stress calculation". For a "thick" workpiece the thickness should be greater than 1.0D (for type A and type B rosettes) and the standard provides the calculation methods for uniform

The range of thicknesses between 0.2D and 1.0D, defined as intermediate thickness, is outside the scope of the ASTM standard. Using a strain gage rosette with a gage circle diameter D = 5.13 mm, the intermediate thickness is identified in the range between 1 and 5.13 mm. Clearly, on varying the diameter of the strain gage

Unfortunately, intermediate thickness is common in several types of engineer-

This limitation in the ASTM standard can be explained by analysis of the behav-

of grids and higher costs.

**Figure 5.**

*a 6-grid rosette.*

calculated stresses.

and non-uniform stress distributions.

circle, also the range of the intermediate thickness varies.

ing applications, as in aerospace, motor sports and energy production.

ior of stress response if a hole is made in an intermediate thickness specimen. In the case of thick workpieces (s > 1.0D) the influence coefficients are independent of the thickness and they can be obtained by an FE model in which the hole

is produced in a virtually semi-infinite body. In the case of thin workpieces

acquire strains.

**68**

*Localized bending caused by hole-drilling in an "intermediate" thickness specimen.*

#### **Figure 7.**

*Calibration coefficients as a function of plate thickness: for aij (left) and for bij, (right) coefficients in the matrices (from Beghini et al. [16]).*

based on 5 different thicknesses (2.7 D, 1.0 D, 0.6 D, 0.3 D, 0.2 D); once the thickness is defined, the displacements are interpolated between the two closest available thickness values.

Blödorn et al. [19] recalculated the ASTM E837 coefficient for blind uniform stress

*Recent Advancements in the Hole-Drilling Strain-Gage Method for Determining Residual Stresses*

chamfer as a new geometrical parameter of the finite element model.

More recently, the generalized integral method based on the influence functions [8, 9] has been enriched with a new database of displacements, which considers the

For a certain value of the ratio between the height of the hole chamfer and the radius of the drilled hole, this methodology allows the correction of calculated stress

**Figure 9** shows the finite element model in which the hole bottom chamfer was

The presence of the hole-bottom chamfer influences the calculation of the stresses. **Figure 10** gives an example of the influence of a hole bottom chamfer on the reconstruction of a pure shear stress distribution. In the first part of the depth of the analysis, it is clearly seen that the chamfer determines an under-estimation of the actual stress, especially in the first depth increments. On the contrary, in the second part of the depth of the analysis, the results show an over-estimation of the

*Finite Element Model used for the evaluation of the calibration coefficients considering the presence of the*

*Residual stresses in the case of pure shear stress, with (dashed line) and without (solid line) the hole-bottom*

using an FEM model with a hole bottom chamfer.

*DOI: http://dx.doi.org/10.5772/intechopen.90392*

for blind holes and non-uniform stress distributions.

simulated to evaluate its influence.

calculated stresses.

**Figure 9.**

**Figure 10.**

*chamfer.*

**71**

*hole-bottom chamfer.*
