**10. Evaluation of uncertainty**

**9. Local plasticity limitation: description and possible correction**

"thick" workpieces and 50% in the case of "thin" workpieces.

*New Challenges in Residual Stress Measurements and Evaluation*

generates larger overall surface strains (**Figure 11**).

stress–strain curve in the plastic region are defined.

strain gage rosettes available on the market.

material under testing and the plasticity factor *f* defined as following:

*Local plasticity areas with low applied loads (left side) or high loads (right side).*

the case of a through hole (50% of σY).

corrections for this error.

plastic strains.

**Figure 11.**

**72**

The ASTM E837 standard reports that satisfactory results can be achieved when measured residual stresses do not exceed about 80% of the yield stress in the case of

The need for these stress limits is explained by the stress concentration generated by the drilled hole. When a hole is drilled on a loaded workpiece, it generates a stress concentration in the area around the hole. The magnitude of the stress concentration depends on several parameters including the diameter of the drilled hole, the load orientation and the distance of the strain gage grids from the hole. If the stress level is high, localized plastic deformation occurs around the hole, which

The hole-drilling method requires that the strain gage grids be placed really close

In any case, "thick" workpieces are less sensitive to the plasticity effect. This is due to the presence of material in the lower part of the blind hole determining a local reinforcement and reducing the stress concentration factor [3]. This explains the higher measurement limit in the case of a blind hole (80% of σY) compared to

Few research studies have been published on this topic to provide possible

The work of Beghini et al. [20] provides a numerical procedure for correcting the effect of local plasticity in the case of a blind hole for uniform stress calculation. To carry out the stress correction, it is necessary that both the yield stress and the

The equivalent stress, corrected to take into account the presence of plasticity *σeq*, is evaluated considering the elastic equivalent stress *σeq*,*<sup>i</sup>*, the yield stress *σ<sup>Y</sup>* of the

> *<sup>f</sup>* <sup>¼</sup> *<sup>σ</sup>eq* � *<sup>σ</sup>eq*,*<sup>i</sup> σ<sup>Y</sup>* � *σeq*,*<sup>i</sup>*

The correction algorithm obviously considers the geometry of the strain gage rosette and therefore the authors provide the calculation coefficients for several

(8)

to the hole. For this reason, if local plasticity occurs, it may be that the strain measured by the gage is the arithmetical sum of the linear elastic strains and the

> The evaluation of uncertainties associated with measurement of residual stresses by the hole-drilling method is a topic that has been little investigated. The evaluation, mainly in the case of non-uniform stress fields, involves a large number of parameters from different sources, including the properties of the materials under testing, the strain readings and the hole execution methods.

> Standard ASTM E837-13a [1] contains only some basic information about precision and bias associated with the hole-drilling measurement method, mainly in the case of uniform stress calculation. In fact, the standard states that the bias associated with a residual stress measurement by the hole-drilling method is less than �10% when dealing with uniform residual stresses. Based on the results of round-robin test programs, the precision (random error) is such as to give a standard deviation of �14 MPa for AISI 1018 carbon steels and a standard deviation of �12 MPa for type AISI 304 stainless steels. The standard also reports that the uncertainties in the case of non-uniform stress measurements are expected to be much larger than for uniform stress measurements.

> One of the first papers on the subject of evaluation of uncertainty was published by Oettel [25] (UNCERT Code of Practice 15). The work proposes an approach for the evaluation of hole-drilling uncertainty in the case of uniform stress fields and takes into account typical errors in the determination of material properties, errors in the measurement of acquired strains, the hole diameter and the influence of calculation coefficients. The code of practice can be applied only to uniform residual stress calculation equations based on ASTM E837-95 and cannot be used with the current version of the ASTM standard [1].

> Scafidi et al. [26] further developed this methodology by applying it to the recent version of the average uniform stress calculation and considering additional parameters, such as the step-by-step drilling depth.

> Regarding evaluation of uncertainty in the case of non-uniform stresses, the first approach was provided by Schajer et al. [24] based on the Integral method. They consider a number of input estimates including the properties of materials (i.e. Young's modulus), strain readings, hole diameter and hole depths. The uncertainty components have statistical normal distributions with zero mean and are independent of each other and each one is linearly combined.

The uncertainties of the measured strains are considered as an input, but fixed for each step.

More recently a new approach was proposed by Peral et al. [27], based on a Monte Carlo analysis of the influence of the main parameters affecting the measurements. The methodology takes into account a higher number of parameters compared to the approach proposed by Schajer. In particular, the uncertainty components due to Poisson's ratio and identification of the zero-depth are also considered. The authors demonstrated that their method is comparable with the approach of Schajer et al. [24], showing generally more conservative results although in good agreement.

SINT Technology recently developed another approach to evaluation of uncertainty, based on the GUM methodology [23], and it is implemented in the calculation software EVAL 7.

Before calculation of uncertainties, all possible systematic errors are corrected, in particular those determined by eccentricity (Section 6), intermediate thickness (Section 7), hole bottom chamfer (Section 8) or local plasticity in the case of uniform stress calculation (Section 9).

The uncertainties determined by the following input parameters are considered: Young's modulus, Poisson's ratio, hole diameter, accuracy of the strain measurement system, zero depth offset error, depth of drilling increments.

This approach can be applied to all available calculation methods including the ASTM standard: clearly, the generalized integral method approach (Section 4) is preferable as it allows several systematic errors to be corrected (Sections 6–8).

The functional relationship *f* that connects the output estimate to the inputs and the measurements can be written, for each calculation step *j*, as:

$$\{\sigma\_{\text{MIN},\text{MAX}}, \emptyset = f\left(\mathbf{E}, \boldsymbol{\nu}, \mathbf{D}\_0, \mathbf{e}\_{1,\text{j}}, \mathbf{e}\_{2,\text{j}}, \mathbf{e}\_{3,\text{j}}, \mathbf{z}\_{\text{j}}, \mathbf{z}\_0\right) \tag{10}$$

where E is the Young's modulus of the material; ν the Poisson's ratio; D0 the diameter of the drilled hole; ε1,j, ε2,j, ε3,j the Readings of the strain gage grids for the step *j*; zj the depth advance for the step *j* and z0 the depth error during the zero-depth determination.

The reading of the strain gage grids, for each calculation step *j* and for each channel *y*, is derived from the following parameters:

$$
\varepsilon\_{1,\circ}, \varepsilon\_{2,\circ}, \varepsilon\_{3,\circ} = f(\mathbf{K\_x}, (\Delta V/V)\_{j,\circ} \tag{11}
$$

*uc*ð Þ¼ *y*

*Parameters used for the uncertainty evaluation and distribution of probability.*

*I* ¼ 1, … , *N* is the number of input quantities.

**Input estimates** *xi*

**Description Sub-input**

E Young's Modulus of

testing

testing

D0 Diameter of the drilled hole

ε1,j, ε2,j, ε3,j Gage factor of the strain gage grids

grid

**Table 3.**

**75**

ν Poisson's ratio of

the materials under

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

the materials under

Electrical output of each strain gage

**estimates**

Kx for each channel *x*

ð Þ *ΔV=V <sup>j</sup>*,*<sup>x</sup>* for each channel *x* for each step *j*

**Distribution Origin**

/ Rectangular Material datasheet

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

/ Rectangular Material datasheet

/ Normal Resolution of the dial gages

Max. error of the dial gages Repeatability of the dial gages

Linearity of the strain gage

Resolution of the strain gage

Noise of the strain gage amplifier

Normal Uncertainty declared on the strain gage datasheet

Normal Class of accuracy of the strain gage amplifier

amplifier

amplifier

mechanical device Max. error between two consecutive steps of the holedrilling mechanical device

mechanical device

standard uncertainty *uC*ð Þ*y* by a coverage factor k:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*u*<sup>2</sup>ð Þ *xI*

*U* ¼ *kuC*ð Þ*y* (13)

*Y* ¼ *y* � *kuC*ð Þ*y* (14)

(12)

*∂f ∂xI* � �<sup>2</sup>

X*<sup>N</sup> I*¼1

zj Depth increment / Normal Resolution of the hole-drilling

z0 Zero-depth error / Rectangular Datasheet of the hole-drilling

where *y* is the measurement result (output estimate); *uC*ð Þ*y* is the combined standard uncertainty for measurement result; *x*<sup>I</sup> the input quantity measurement (input estimate); *u x*ð Þ*<sup>I</sup>* is the standard uncertainty for each input quantity;

Finally, the expanded uncertainty *U* is obtained by multiplying the combined

The advantage of this method is the capability to numerically evaluate, for each parameter and for each calculation depth, the first derivatives of the functional

s

The result of a measurement is then conveniently expressed as:

where Kx is the Gage Factors of the strain gage for each grid *x*; ð Þ *ΔV=V <sup>j</sup>*,*<sup>x</sup>* is the electrical signal output for each channel *x* and for each step *j*.

The uncertainty on the electrical output ð Þ *ΔV=V <sup>j</sup>*,*<sup>x</sup>* depends on the strain gage amplifier technical specifications, such as class of accuracy or resolution, linearity and noise to signal ratio. This information can be obtained from the technical datasheet of the strain gage amplifier.

Also the uncertainty on the measurement of the drilled hole D0 and the depth increments zj, for each *j* step, are evaluated starting from the technical specification respectively of the dial gages used for the hole measurements and the mechanical drilling unit that makes the hole.

The uncertainty component due to the temperature variation during testing is considered negligible as the strain gages are self-compensated and a three-wire halfbridge connection is adopted to minimize the effect of temperature on the cables. Also the uncertainties due to the Influence Functions are considered negligible. **Table 3** shows the typical input parameters taken into account for the evaluation of the uncertainty, along with the type of statistical distribution.

Assuming that all the input quantities are independent, the combined standard uncertainty, for each calculation step *j*, is given by:


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

#### **Table 3.**

More recently a new approach was proposed by Peral et al. [27], based on a Monte Carlo analysis of the influence of the main parameters affecting the measurements. The methodology takes into account a higher number of parameters compared to the approach proposed by Schajer. In particular, the uncertainty components due to Poisson's ratio and identification of the zero-depth are also considered. The authors demonstrated that their method is comparable with the approach of Schajer et al. [24], showing generally more conservative results although in good agreement.

SINT Technology recently developed another approach to evaluation of uncertainty, based on the GUM methodology [23], and it is implemented in the calcula-

Before calculation of uncertainties, all possible systematic errors are corrected, in particular those determined by eccentricity (Section 6), intermediate thickness (Section 7), hole bottom chamfer (Section 8) or local plasticity in the case of

The uncertainties determined by the following input parameters are considered: Young's modulus, Poisson's ratio, hole diameter, accuracy of the strain measure-

The functional relationship *f* that connects the output estimate to the inputs and

(10)

ε1,j, ε2,j, ε3,j ¼ *f*ðKx,ð Þ *ΔV=V <sup>j</sup>*,*<sup>x</sup>* (11)

This approach can be applied to all available calculation methods including the ASTM standard: clearly, the generalized integral method approach (Section 4) is preferable as it allows several systematic errors to be corrected (Sections 6–8).

*σMIN*,*MAX*, β ¼ *f* E, ν, D0, ε1,j, ε2,j, ε3,j, zj, z0

where E is the Young's modulus of the material; ν the Poisson's ratio; D0 the diameter of the drilled hole; ε1,j, ε2,j, ε3,j the Readings of the strain gage grids for the step *j*; zj the depth advance for the step *j* and z0 the depth error during the

The reading of the strain gage grids, for each calculation step *j* and for each

where Kx is the Gage Factors of the strain gage for each grid *x*; ð Þ *ΔV=V <sup>j</sup>*,*<sup>x</sup>* is the

The uncertainty on the electrical output ð Þ *ΔV=V <sup>j</sup>*,*<sup>x</sup>* depends on the strain gage amplifier technical specifications, such as class of accuracy or resolution, linearity and noise to signal ratio. This information can be obtained from the technical

Also the uncertainty on the measurement of the drilled hole D0 and the depth increments zj, for each *j* step, are evaluated starting from the technical specification respectively of the dial gages used for the hole measurements and the mechanical

The uncertainty component due to the temperature variation during testing is considered negligible as the strain gages are self-compensated and a three-wire halfbridge connection is adopted to minimize the effect of temperature on the cables. Also the uncertainties due to the Influence Functions are considered negligible. **Table 3** shows the typical input parameters taken into account for the evaluation of

Assuming that all the input quantities are independent, the combined standard

ment system, zero depth offset error, depth of drilling increments.

*New Challenges in Residual Stress Measurements and Evaluation*

the measurements can be written, for each calculation step *j*, as:

channel *y*, is derived from the following parameters:

electrical signal output for each channel *x* and for each step *j*.

the uncertainty, along with the type of statistical distribution.

uncertainty, for each calculation step *j*, is given by:

tion software EVAL 7.

zero-depth determination.

datasheet of the strain gage amplifier.

drilling unit that makes the hole.

**74**

uniform stress calculation (Section 9).

*Parameters used for the uncertainty evaluation and distribution of probability.*

$$u\_c(\mathbf{y}) = \sqrt{\sum\_{I=1}^{N} \left(\frac{\partial f}{\partial \mathbf{x}\_I}\right)^2 u^2(\mathbf{x}\_I)}\tag{12}$$

where *y* is the measurement result (output estimate); *uC*ð Þ*y* is the combined standard uncertainty for measurement result; *x*<sup>I</sup> the input quantity measurement (input estimate); *u x*ð Þ*<sup>I</sup>* is the standard uncertainty for each input quantity; *I* ¼ 1, … , *N* is the number of input quantities.

Finally, the expanded uncertainty *U* is obtained by multiplying the combined standard uncertainty *uC*ð Þ*y* by a coverage factor k:

$$U = k u\_C(\mathcal{y})\tag{13}$$

The result of a measurement is then conveniently expressed as:

$$Y = \mathbf{y} \pm k u\_C(\mathbf{y})\tag{14}$$

The advantage of this method is the capability to numerically evaluate, for each parameter and for each calculation depth, the first derivatives of the functional

relation *f* (sensitivity coefficients) which, in the formula (12), multiply the standard uncertainty square *u x*ð Þ<sup>i</sup> of each input estimate *x*i.

where *b* is the distance between the load axis and the rosette strain gage center, *wh2* is the width and height of the beam cross section, and *F* is the load imposed by

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

A known load was used for determining the properties of the material. In fact, the elasticity parameters (Young's modulus E and Poisson's ratio ν) of the material were measured before drilling by applying a preliminary bending load before

Grid 1 of each strain gage should be aligned with the beam axis. The manual strain gage installation unavoidably introduces a misalignment. However, the angle between grid 1 and the beam axis can be found from Eq. 16 (accurate approxima-

<sup>1</sup> ð Þ� <sup>0</sup> <sup>2</sup>*ε<sup>F</sup>*

*εF* <sup>1</sup> ð Þ� 0 *ε<sup>F</sup>*

*εRS <sup>i</sup> zj*

> *<sup>i</sup> zj* � *<sup>ε</sup><sup>i</sup> zj*

The following testing conditions were adopted during the measurements.

The following parameters were then used for the stress calculation and for the

• Uncertainty on the maximum electrical output (*ΔV/V*): �0.50 μm/m

The measured strains need to be decoupled in order to deduce the relaxed strain due to the bending stress. The relaxed strains due to the residual stresses and the

<sup>¼</sup> *<sup>ε</sup><sup>i</sup> zj*

*<sup>i</sup>* ð Þ 0 needs to be subtracted in the second member of Eq. 17 since the relaxed strains are defined as the effect of introduction of the drilled hole, therefore they need to be zero at zero depth. Finally, the experimental data are the bending

� *<sup>ε</sup><sup>F</sup>*

*<sup>i</sup> zj* .

<sup>2</sup> ð Þþ <sup>0</sup> *<sup>ε</sup><sup>F</sup>*

<sup>3</sup> ð Þ 0

<sup>3</sup> ð Þ <sup>0</sup> (16)

<sup>1</sup> ð Þ <sup>0</sup> (17)

*<sup>γ</sup>* <sup>¼</sup> <sup>1</sup> 2 � *εF*

relaxed strains due to the bending stresses are obtained as:

*εBe <sup>i</sup> zj* <sup>¼</sup> *<sup>ε</sup><sup>F</sup>*

relaxed strains as a function of hole depth increments *εBe*

the pneumatic actuator.

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

tion for small values of γ):

Strain *ε<sup>F</sup>*

**12. Test results and analysis**

• Surface bending stress (*σBe*): 24.8 MPa

• Strain gage rosette: CEA-062UM-120

• Measured eccentricity radius: 0.02 mm

• Young's modulus (*E*): 71000 � 3550 MPa

• Measured eccentricity angle: 135°

• Poisson's ratio (*ν*): 0.33 � 0.01

• Gage factor uncertainty (*K*): 1%

• Hole diameter (*D0*): 1.88 � 0.01 mm

uncertainty evaluation:

**77**

drilling.

This calculation procedure, which is implemented in the EVAL 7 software, requires the execution of a high number of stress calculations for the uncertainty evaluation related to hole drilling measurements.

In particular, considering a measurement carried out according to the ASTM standard using 20 calculation depths, the uncertainty evaluation requires the repetition and therefore the combination of the results obtained with 206 different stress calculations.
