Estimation formula for residual stress from the blind-hole drilling method

Abstract

In this study, the accuracy of blind-hole method on weld residual stress estimation is investigated. A modified parameter group has also proposed to improve the accuracy. The thermal-elastic-plastic finite element model is employed to build up the residual stress distribution and the blind-hole process. The MSC Marc finite element software package is used to simulate the welding process and the welding residual stress and strain distributions around the weld of two inconel 690 alloy plates filled with I-52 GTAW filler. Then the process of the traditional blind hole is simulated by employing the inactive elements. The data of the residual strain variations of strain gages located around the blind hole is introduced into the blind-hole method to estimate the original residual stress components at the hole center. The effects of drilling depth, drilling size, gage radius, gage position, and the distance on the accuracy of estimated residual stress have also been studied and discussed. Based on the residual stress components simulated from the welding process, a modified stress parameter group has also been proposed to improve the accuracy of blind-hole method. Numerical results indicate that the accuracy of estimated residual stress can be improved significantly by employing the proposed blind-hole parameters.

Keywords

Blind-hole drilling methods residual stress finite element method residual strain calibration coefficient

Introduction

The hole-drilling strain-gage method can be divided into through-hole drilling and blind-hole drilling (BHD) methods. The through-hole drilling method involves directly drilling holes through a workpiece and observing the elastic strain energy release. The BHD method involves drilling a hole to a depth comparable with the hole size. Because the BHD method only generates local fractures, and the small hole drilled typically does not affect the normal usage of mechanical objects, the BHD method has been widely applied in industrial practice.

During an actual strain measurement process, measuring precise strain values is difficult and requires multiple measurements, hole-drilling method is the most to be used.¹ To employ the advanced computational capacity and enhanced software functions of computers, a finite element software package MSC Marc was employed to generate a residual stress field in a simulated engineering processing procedure. Subsequently, the BHD method was applied to determine the strain value around the hole and theoretically calculate the residual stress. On the basis of the simulation results, applicability of the BHD method was identified. The effect on the result of BHD parameters, such as changes in hole depth, distance from the welding bead center, initial position of the strain gage, gage radius ratio, and hole radius, was observed. A numerical analysis was applied to compare the BHD parameters, and the results were referenced to develop a more accurate method to measure residual stress.

P Gomes² presents the results of the residual strain and the stainless steel AISI 304. Steinzig³ showed that accurate results can be achieved without ultra-high drill-rotation speed in aluminum and stainless steel. MR Viotti⁴ found the influence of the feed and rotational speed on the hole walls, and the bottoms were evaluated for A36 steel and AISI 304L stainless steel.

BHD theory

This study explored the use of finite element software to simulate the residual stress generated on metal plates following heating and condensation. In addition, the deactivation software function was applied to remove BHD element simulation to estimate residual stress.

Theory

Because a hole with a radius of r_o on an infinite plane slab is affected by a tensile plane stress, the stress state around the hole, according to elasticity explanation,⁵ can be considered as superposition of Case 1 and Case 2, also shown respectively in Figure 1(a) and (b). The stress distribution in Case 1 can be expressed as follows

σ_{r}' (r) = \frac{σ_{x}}{2} [1 - {(\frac{r_{o}}{r})}^{2}]

(1)

σ_{β}' (r) = \frac{σ_{x}}{2} [1 + {(\frac{r_{o}}{r})}^{2}]

(2)

{τ_{γ β}}^{'} (r) = 0

(3)

The stress distribution in Case 2 can be expressed as follows

σ_{r}' (r, β) = \frac{σ_{x}}{2} [1 - 4 {(\frac{r_{o}}{r})}^{2} + 3 {(\frac{r_{o}}{r})}^{4}] \cos 2 β

(4)

σ_{β}' (r, β) = - \frac{σ_{x}}{2} [1 + 3 {(\frac{r_{o}}{r})}^{4}] \cos 2 β

(5)

τ_{r β}' (r, β) = - \frac{σ_{x}}{2} [1 + 2 {(\frac{r_{o}}{r})}^{2} - 3 {(\frac{r_{o}}{r})}^{4}] s i n 2 β

(6)

where $σ_{x}$ is a tensile plane stress, ${σ^{'}}_{r}$ is a stress component along the r axis, $σ'_{β}$ is a stress component perpendicular to the r axis, r₀ is the hole radius, r is the strain-gage radius, and $β$ is the angle between the position vector of point P and $σ_{x}$ direction.

Figure 1.

Circumferential stress around the hole of a drilled infinite plane slab affected by a tensile plane stress.

σ_{r}^{'} (r, β) = \frac{σ_{x}}{2} [1 - {(\frac{r_{o}}{r})}^{2}] + \frac{σ_{x}}{2} [1 - 4 {(\frac{r_{o}}{r})}^{2} + 3 {(\frac{r_{o}}{r})}^{4}] c o s 2 β

(7)

σ_{β}^{'} (r, β) = \frac{σ_{x}}{2} [1 + {(\frac{r_{o}}{r})}^{2}] - \frac{σ_{x}}{2} [1 + 3 {(\frac{r_{o}}{r})}^{4}] c o s 2 β

(8)

{τ_{r β}}^{'} (r, β) = - \frac{σ_{x}}{2} [1 + 2 {(\frac{r_{o}}{r})}^{2} - 3 {(\frac{r_{o}}{r})}^{4}] \sin 2 β

(9)

The stress of drilling point $P (r, β)$

σ_{r} = \frac{σ_{x}}{2} (1 + \cos 2 β)

(10)

σ_{β} = \frac{σ_{x}}{2} (1 - \cos 2 β)

(11)

τ_{r β} = - \frac{σ_{x}}{2} \sin 2 β

(12)

Comparing the stress of the point $P (r, β)$ before and after drilling can yield a calculation of the stress relaxation

Δ σ_{r} = {σ_{r}}^{'} - σ_{r}

(13)

Δ σ_{β} = {σ_{β}}^{'} - σ_{β}

(14)

Substituting equations (7) and (10) into equation (13) results to equation (15)

Δ σ_{r} = \frac{σ_{x}}{2} [- {(\frac{r_{o}}{r})}^{2}] + \frac{σ_{x}}{2} [- 4 {(\frac{r_{o}}{r})}^{2} + 3 {(\frac{r_{o}}{r})}^{4}] \cos 2 β

(15)

Substituting equations (8) and (11) into equation (14) results to equation (16)

Δ σ_{β} = \frac{σ_{x}}{2} {(\frac{r_{o}}{r})}^{2} - \frac{3 σ_{x}}{2} {(\frac{r_{o}}{r})}^{4} \cos 2 β

(16)

According to Kabiri,⁶ the radial strain change is substantially greater than the circumferential strain change. When $Δ ε_{β}$ is ignored, the resulting error is far lower than that caused by other measurement techniques. Thus, the effect of the circumferential strain is omitted during calculation. When the thin plate under force is of a homogeneous and isotropic material and is linear-elastic, biaxial Hooke’s law is applied to connect the stress–strain relationship

Δ ε_{r} = \frac{1}{E} (Δ σ_{r} - v Δ σ_{β})

(17)

Substituting equations (15) and (16) into equation (17) gives equation (18)

Δ ε_{r} = σ_{x} (A + Bcos 2 β)

(18)

A = - \frac{1 + v}{2 E} {(\frac{r_{o}}{r})}^{2}

(19)

B = - \frac{1 + v}{2 E} [(\frac{4}{1 + v}) {(\frac{r_{o}}{r})}^{2} - 3 {(\frac{r_{o}}{r})}^{4}]

(20)

Δ ε_{r} = σ_{x} (A + Bcos 2 β) + σ_{y} (A - Bcos 2 β)

(21)

Equation (21) shows that three unknown values $σ_{x}$ , $σ_{y}$ , and $β$ are present in the equation; in other words, strain measurement values along three or more directions are necessary to solve the equation. The distance from the center is r, and the angles between the position vector and stress direction are $β$ , $β + \emptyset$ , and $β + ψ$ for points $P (r, β)$ , $Q (r, β + \emptyset)$ , and $R (r, β + ψ)$ , respectively. Substituting the three points into equation (21) determines the strain change of the points

{(Δ ε_{r})}_{P} = [A + Bcos 2 β] σ_{x} + [A - Bcos 2 β] σ_{y}

(22)

{(Δ ε_{r})}_{Q} = [A + Bcos 2 (β + \emptyset)] σ_{x} + [A - Bcos 2 β] σ_{y}

(23)

{(Δ ε_{r})}_{R} = [A + Bcos 2 (β + ψ)] σ_{x} + [A - Bcos 2 (β + ψ)] σ_{y}

(24)

Solving the simultaneous equations of (22), (23), and (24) yields the following equations

σ_{x} = \frac{1}{2 A} \frac{{(Δ ε_{r})}_{R} \cos 2 β - {(Δ ε_{r})}_{P} \cos 2 (β + ψ)}{\cos 2 β - \cos 2 (β + ψ)} + \frac{1}{2 B} \frac{{(Δ ε_{r})}_{P} - {(Δ ε_{r})}_{R}}{\cos 2 β - \cos 2 (β + ψ)}

(25)

σ_{y} = \frac{1}{2 A} \frac{{(Δ ε_{r})}_{R} \cos 2 β - {(Δ ε_{r})}_{P} \cos 2 (β + ψ)}{\cos 2 β - \cos 2 (β + ψ)} - \frac{1}{2 B} \frac{{(Δ ε_{r})}_{P} - {(Δ ε_{r})}_{R}}{\cos 2 β - \cos 2 (β + ψ)}

(26)

β = \frac{1}{2} ta n^{- 1} {\frac{[\cos 2 ψ [{(Δ ε_{r})}_{Q} - {(Δ ε_{r})}_{P}] + \cos 2 φ [{(Δ ε_{r})}_{P} - {(Δ ε_{r})}_{R}] + [{(Δ ε_{r})}_{R} - {(Δ ε_{r})}_{Q}]]}{[\sin 2 ψ [{(Δ ε_{r})}_{Q} - {(Δ ε_{r})}_{P}] + \sin 2 φ [{(Δ ε_{r})}_{P} - {(Δ ε_{r})}_{R}]]}}

(27)

During the deduction process, the angle is given a positive sign when it is in a clockwise direction. Thus, when a commonly used principal stress coordinate relationship is integrated into equation (27), the positive sign is instead given to the counterclockwise direction.

In reality, BHD experiments are conducted on slabs with a finite width and the measured strain value is the average strain value of the area covered by the strain gage, instead of a point. Rendler and Vigness⁷ asserted that the theory of the through-hole drilling method is applicable to the BHD method. However, the A and B coefficients in equations (19) and (20) must be corrected to calibration coefficients $\bar{A}$ and $\bar{B}$ , obtained through a calibration experiment. Schajer⁸ applied a numerical method to solve the problem of the strain-gage coverage area to determine general solutions for calibration coefficients $\bar{A}$ and $\bar{B}$

\bar{A} = - \frac{1 + v}{2 E} \bar{a}

(28)

\bar{B} = - \frac{1}{2 E} \bar{b}

(29)

BHD parameter setting

Figure 2 shows the geometric parameters and relevant sign definition required for estimating residual stress based on the BHD method. The initial position of the first strain gage is $θ$ , $φ$ is the angle between the first and second strain gages and $ψ$ is the angle between the first and third strain gages. In accordance with the Kirsch formula,⁹ the angles are positive in the clockwise direction. The hole radius is r₀, r is the radial distance from the strain gage to the hole center, t is the slab thickness, h is the hole depth, w is the distance between the strain gage and the slab edge, and L is the distance between the hole center and the welding bead. Some parameters were non-dimensionalized to facilitate subsequent analysis. The hole depth is $h / t$ , $L / t$ is the distance from the hole, and r₀/r is the gage radius.

Figure 2.

Geometric parameters.

Heat source setting

For heat source, scholars used TEM₀₀ modal in the past. The highest laser power density concentrates on the center, gradually decreases with increasing radius, formed Gaussian distribution, so it is also called Gaussian laser. TEM₀₀ laser power density as followed

E_{e} (r_{e}) = P \cdot Φ (r_{e})

(30)

where $r_{e}$ is the distance of laser center, P is output of laser power, $Φ$ is Gaussian distribution function (fundamental unit is $L^{- 2}$ ). Gaussian distribution functions as followed

Φ (r_{e}) = \frac{1}{2 π s^{2}} \exp (- \frac{r_{e}^{2}}{2 s^{2}})

(31)

where s is standard deviation. Define R_e is the radius of spot size, 95% power covered the scope

\int_{0}^{2 π} \int_{0}^{R_{e}} \frac{1}{2 π s^{2}} ex p^{- \frac{{r_{e}}^{2}}{2 s^{2}} r_{e} d r_{e} d θ} = 0.95

(32)

s = \frac{R_{e}}{\sqrt{2 \ln (20)}} \approx \frac{R_{e}}{\sqrt{6}}

(33)

Substituting equation (33) into (32) forms equation (34)

E_{e} (r_{e}) = \frac{3 P}{π R_{e}^{2}} \exp (- \frac{3 r_{e}^{2}}{R_{e}^{2}})

(34)

In 1984, Goldak et al.¹⁰ based on two dimensions Gaussian heat source proposed a double-ellipsoidal Gaussian heat source model. Model is divided by electric arc center as shown in Figure 3, the model of x, y, z distribution power decrease by Gaussian function.

Figure 3.

Double-ellipsoidal heat model.

Finite analysis by double-ellipsoidal Gaussian heat source presented not only the bath during arc welding short-former long-latter, but also the heat source distribution of depth.

Double-ellipsoidal Gaussian heat source model have two part functions, function of former half is

E_{e} (x . y . z) = \frac{6 \sqrt{3} f_{1} P}{π a_{1} bc \sqrt{π}} \exp (- 3 \frac{x^{2}}{a_{1}^{2}}) \exp (- 3 \frac{y^{2}}{b^{2}}) \exp (- 3 \frac{z^{2}}{c^{2}})

(35)

and function of latter half is

E_{e} (x, y, z) = \frac{6 \sqrt{2} f_{2} P}{π a_{2} b c \sqrt{π}} e x p (- 3 \frac{x^{2}}{a_{2}^{2}}) e x p (- 3 \frac{y^{2}}{b^{2}}) e x p (- 3 \frac{z^{2}}{c^{2}})

(36)

$f_{1}$ , $f_{2}$ is respected heat distribution function by former and latter ellipsoidal, adding of them is two. Welding setting by the heat source model, laser output P is replaced by welding power Q

Q = η_{w} IV

(37)

where η_w is the welding heat efficiency, I is the welding current, and V is the electric arc voltage.

Finite element model

In accordance with Lee,¹¹ two metal plates of inconel alloy 690 of 100-mm length, 70-mm width, and 3-mm thickness were welded using V Groove butt welding, included angle 90°, bottom gap 2 mm, distance from bottom 1 mm. A finite element model was established with the dimensions depicted in Figure 4. Solder used INCONEL Filler Metal 52 (I-52) for welding. I-52 has better melting flow, resists pyrolysis and solidification crack ability. The alloy composition is similar to inconel alloy 690.¹² The distance between each hole center and welding bead central line is listed in Table 1. Thermomechanical coupling was applied to reduce the consumption of calculation resources. A semi-symmetric model was applied for the analysis, as shown in Figure 5. The material properties of Inconel alloy 690 and I-52 are shown in Figure 6. The simulated boundary conditions of welding thermal residual stress were set as follows.

Figure 4.

Dimensions of welded metal plate.

Table 1.

Distance between hole center and welding bead central line.

Position	Distance L, (mm)
$O_{1}$	16.86
$O_{2}$	21.87
$O_{3}$	36
$O_{4}$	45
$O_{5}$	60

Figure 5.

Diagram of element cutting.

Figure 6.

Material properties of Inconel alloy 690 and I-52: (a) Young’s modulus, (b) yield strength, (c) thermal expansion, (d) thermal conductivity, and (e) specific heat.^11,12

During the welding process, the environment temperature was set as 20°C and the solder melting point was 1350°C. At the first welding bead, the welding velocity, welding bead length, and welding bead height were 1.4 mm/s, 70 mm, and 2 mm respectively. At the second welding bead, the welding velocity, welding bead length, and welding bead height were 1.8 mm/s, 70 mm, and 1.6 mm, respectively. The two plane slabs dissipated heat through natural convection as the temperature reduced to room temperature. In the geometric shape of the proposed model, the convection coefficient was set as 13.5 W/(m² K). The deactivation element command of the applied software program was employed to remove blind-hole elements to attain the re-equilibrium of the residual stress field.

Non-dimensional coefficient correction

In accordance with the theory and simulation steps described in previous sections, this section investigates the difference between the residual stress field of metal plate welding obtained in a finite element simulation and the residual stress field estimated using the American society for Testing and Materials (ASTM) E 837-01 norms. The least squares error method was applied to correct the coefficients proposed by the norms.

Figures 7 and 8 show the values estimated using ASTM norms at the position, respectively. The estimated values could not simultaneously satisfy individual optimal precisions.

Figure 7.

ASTM-based estimation of $σ_{1}$ at the $O_{3}$ position.

Figure 8.

ASTM-based estimation of $σ_{2}$ at the $O_{3}$ position.

Thus, the least square error method was applied to correct the non-dimensional coefficients proposed by ASTM norms, thereby determining the non-dimensional parameters that improve the estimation accuracy. The stress of each reference point calculated using the finite element method is depicted in Table 2. The domains of parameter changes are tabulated in Table 3.

Table 2.

Simulation stress values of each reference point.

Position	Principal stress
	$σ_{1}$ (MPa)	$σ_{2}$ (MPa)
$O_{1}$	129.0	–110.0
$O_{2}$	65.1	–202.0
$O_{3}$	–32.1	–110.0
$O_{4}$	–33.9	–41.4
$O_{5}$	24.0	–23.6

Table 3.

Corrected non-dimensional coefficient ${\bar{a}}^{*}$ .

$h / 2 r$	Gage radius $r_{o} / r$
	0.3	0.35	0.4	0.45	0.5
0.1	0.057	0.074	0.096	0.128	0.160
0.15	0.082	0.110	0.146	0.195	0.246
0.2	0.108	0.138	0.177	0.235	0.298
0.25	0.121	0.157	0.204	0.266	0.329
0.3	0.128	0.170	0.219	0.289	0.360
0.35	0.133	0.177	0.240	0.303	0.367

The correction coefficients ${\bar{a}}^{*}$ and ${\bar{b}}^{*}$ , in place of the original non-dimensional coefficients $\bar{a}$ and $\bar{b}$ , were assumed and used to estimate the maximum principal stress ${\hat{σ}}_{1}$ . The correction coefficients used in equations (28) and (29) can be modified into equations (38) and (39)

{\bar{A}}^{*} = - \frac{1 + v}{2 E} {\bar{a}}^{*}

(38)

{\bar{B}}^{*} = - \frac{1}{2 E} {\bar{b}}^{*}

(39)

Substituting equations (38) and (39) into equations (25) and (26) and defining the objective function $C_{1} (σ_{1}, {\hat{σ}}_{1})$ as featuring the calculation of the error sum of squares of the maximum principal stress yields the following equation

C_{1} (σ_{1}, {\bar{σ}}_{1}^{*}) = \sum_{i = 1}^{n} {[\frac{{({\bar{σ}}_{1}^{*})}_{i} - {(σ_{1})}_{i}}{{(σ_{1})}_{i}}]}^{2}

(40)

A partial derivative of the regression coefficients ${\bar{a}}^{*}$ and ${\bar{b}}^{*}$ can be rearranged to determine the corrected non-dimensional coefficients, as shown in Tables 3 and 4. Similarly, if ${\bar{c}}^{*}$ and ${\bar{d}}^{*}$ are the correction coefficients used to estimate the minimum principal stress ${\hat{σ}}_{2}$ , the principal stress estimation equation, error relationship, and objective function can all then be obtained

Table 4.

Corrected non-dimensional coefficient ${\bar{b}}^{*}$ .

$h / 2 r$	Gage radius $r_{o} / r$
	0.3	0.35	0.4	0.45	0.5
0.1	0.127	0.168	0.212	0.272	0.332
0.15	0.196	0.265	0.346	0.446	0.544
0.2	0.266	0.351	0.448	0.584	0.720
0.25	0.328	0.428	0.549	0.706	0.861
0.3	0.372	0.498	0.643	0.827	1.001
0.35	0.408	0.556	0.750	0.946	1.141

C_{2} (σ_{2}, {\hat{σ}}_{2}^{*}) = \sum_{i = 1}^{n} {[\frac{{({\bar{σ}}_{2}^{*})}_{i} - {(σ_{2})}_{i}}{{(σ_{2})}_{i}}]}^{2}

(41)

The resulting ${\bar{c}}^{*}$ and ${\bar{d}}^{*}$ coefficients are tabulated in Tables 5 and 6. The non-dimensional coefficients shown in Tables 2 –4 were substituted into the Kirsch formula. These results were then compared with the results obtained from the ASTM E 837-01 norms.

Table 5.

Corrected non-dimensional coefficient ${\bar{c}}^{*}$ .

$h / 2 r$	Gage radius $r_{o} / r$
	0.3	0.35	0.4	0.45	0.5
0.1	0.056	0.072	0.093	0.123	0.153
0.15	0.081	0.105	0.137	0.180	0.225
0.2	0.106	0.131	0.163	0.211	0.261
0.25	0.115	0.144	0.184	0.232	0.278
0.3	0.119	0.152	0.189	0.241	0.294
0.35	0.122	0.152	0.196	0.240	0.283

Table 6.

Corrected non-dimensional coefficient ${\bar{d}}^{*}$ .

$h / 2 r$	Gage radius $r_{o} / r$
	0.3	0.35	0.4	0.45	0.5
0.1	–0.118	–0.154	–0.194	–0.249	–0.301
0.15	–0.181	–0.237	–0.303	–0.382	–0.461
0.2	–0.245	–0.309	–0.380	–0.474	–0.567
0.25	–0.290	–0.363	–0.450	–0.548	–0.643
0.3	–0.322	–0.408	–0.496	–0.602	–0.700
0.35	–0.355	–0.434	–0.538	–0.641	–0.744

Parameter discussion

The results of corrected ASTM non-dimensional coefficients show that the corrected non-dimensional coefficients do not necessarily generate more accurate results in all conditions than ASTM coefficients. Thus, all parameters were included for discussion. Moreover, the optimal condition to use corrected non-dimensional parameters in BHD testing was proposed to achieve accurate estimation of the welding residual stress.

The reference value of each parameter was set as follows: hole depth = 0.25 mm, distance from the welding bead center = 12 mm, strain-gage angle = 0°, hole radius = 1 mm, and gage radius = 0.4 mm. The domains of parameter changes were 0.1–0.55 mm for the hole depth, 5.62–15 mm for the distance from the welding bead center, 0°– $3 π / 2$ for the strain-gage angle, 0.5–1.5 mm for hole radius, and 0.33–0.5 for the gage radius.

The parameters and the domains of their changes are tabulated in Table 7. To investigate the blind-hole depth, the reference points with different distances from the welding bead center mentioned in the preceding two sections were applied for an analysis. In each analysis, one parameter, such as distance, strain-gage angle, hole radius, or gage radius, was changed, while the other parameters remained as the reference values. The accuracy of the analysis results were compared with those yielded from using the ASTM E 837-01 norms.

Table 7.

Parameter setting.

Parameters	Reference value (mm)	Domain (mm)
Hole depth $(h / t)$	0.25	0.1–0.55
Distance $(L / t)$	12	5.62–20
Gage location $(θ)$	0	0– $3 π / 2$
Radius $(r_{0})$	1	0.5–1.5
Gage radius $(r_{0} / r)$	0.4	0.33–0.5

Figure 9 shows the relationship between the stress percentage error at each reference point and the hole depth. At the reference point with a distance of 20 mm from the welding bead center, the estimation error was considerably higher than it was at other positions; thus, the distance of 20 mm was not included as a recommended domain. With 0.25 and 0.4 mm hole depths, the $σ_{1}$ of other reference points generated stress errors between 13.8% and –14.16%. The stress error of $σ_{2}$ ranged from 3.6% to –5.22% when the hole depth was 0.25 mm, which was superior to the error range between 4.43% and –6.86% when the hole depth was 0.4 mm.

Figure 9.

Relationship between the corrected estimation stress and welding bead center position.

As shown in Figure 9, corrected coefficients were also applied to estimate $σ_{1}$ . After the point with a distance of 20 mm was omitted, the stress error range at all reference points in Figure 9 was between 13.8% and –14.16%. The stress error range estimated on the basis of the ASTM norms was between 43.37% and –34.97%. During estimation of σ₂, the accuracy of corrected coefficient results was slightly lower than that of the ASTM norm estimation results at positions near the welding bead center. By contrast, the corrected accuracy was higher than that of the ASTM norm estimation results at reference points away from the welding bead center. The analysis results revealed that within a 5.62–15 mm distance from the welding bead center, the corrected coefficients were more accurate than the ASTM norm coefficients in estimating residual stress. The correction effect was more prominent when the ASTM norm estimation error was higher.

The stress estimation errors based on the corrected coefficients and radial strain changes of different initial strain-gage angles were compared with the ASTM norm estimation errors, as depicted in Figure 10, As shown in Figure 10(a), which depicts the $σ_{1}$ stress errors, when $θ$ was zero, the stress error changed from –34.97% to –14.16%; when $θ = 3 π / 2$ , the estimation error was reduced to –6.1%. The $σ_{2}$ errors estimated using corrected coefficients at different initial strain-gage angles were higher than those estimated using ASTM norm coefficients. When $θ = 3 π / 2$ , the estimation error was 0.27%.

Figure 10.

Relationship between the corrected estimation stress and the strain-gage angle: (a) σ₁ and (b) σ₂.

These results indicate that estimating welding residual stress by using corrected coefficients at different strain-gage angles and distances from the welding bead center can limit the estimation error within 15%. Compared with the estimation results yielded using ASTM norms, the corrected coefficients effectively improve the accuracy of the BHD method in estimating the residual stress in welding.

Figure 11 depicts the comparison between accuracies of hole radii estimated using the corrected coefficients and ASTM norm coefficients. In estimation of $σ_{1}$ stress errors, the accuracy improved regardless of increasing the hole radius to 1.5 mm or decreasing it to 0.5 mm; the error was reduced from –14.16% to 0.57% and –3.68%, respectively. In estimation of $σ_{2}$ , the corrected coefficients were applied to estimate the results. When the hole radius increased, the error value increased to 10.06%; when the hole radius decreased, the stress error became 0.84%.

Figure 11.

Relationship between the corrected estimation stress and hole radius. (a) σ₁ and (b) σ₂.

Figure 12 depicts an accuracy comparison of gage radius ratios between the corrected and ASTM norm estimation results. When the hole radius was 1 mm, the gage radius ratio had little effect on estimation accuracy. After the gage radius ratio increased from 0.33 to 0.5 mm, the $σ_{1}$ stress error was changed from –13.98% to –16.56% and the $σ_{2}$ stress error was changed from –0.26% to –1.01%. Both estimation results formed approximately horizontal lines. When the hole radius was reduced to 0.5 mm, the optimal estimation accuracy for $σ_{1}$ and $σ_{2}$ were –3.68% and 0.84%, respectively.

Figure 12.

Relationship between corrected estimation stress and gage radius ratios. (a) σ₁ and (b) σ₂.

The analysis results showed that to lower the fracture level on the specimen while maintaining estimation accuracy, the hole radius should be selected from a range between 0.5 and 1 mm. In accordance with the recommended hole radius range, the optimal domain for the gage radius ratio was 0.4 mm. Finally, the recommended domains of the parameters, such as hole depth, distance from the welding bead center, initial strain-gage angle, hole radius, and gage radius ratio are tabulated in Table 8.

Table 8.

Recommended domain for each parameter.

Parameters	Domain (mm)
Hole depth $(h / t)$	0.25
Distance $(L / t)$	12
Gage location $(θ)$	$3 π / 2$
Radius $(r_{0})$	0.5 mm
Gage radius $(r_{0} / r)$	0.4

Conclusion

A finite element analysis model for simulating and analyzing the welding-induced residual stress field and BHD applications was proposed in this study. By comparing the estimation of principal stress and the finite element results, combined with applying the least squares error method, new non-dimensional coefficients were developed. The recommended domains for each parameter in this study revealed that the proposed calculation methods generated more accurate results than did the ASTM E 837-01 norms.

A finite element analysis model for simulating and analyzing the welding-induced residual stress field and BHD applications was proposed in this study.

The study also used the blind-hole specification which provides the dimensionless coefficients $\bar{a}$ , $\bar{b}$ , calculated with the node strain variable, verified blind-hole method is suitable for finite element simulation. Comparative estimates of principal stress and finite element result found that there is a certain amount of error.

Measurement of residual stress, there are significant impact stress-affected zone on the results, including distance, blind-hole depth, pore size, and location parameters such as strain gage.

A new dimensionless coefficients ${\bar{a}}^{*}$ , ${\bar{b}}^{*}$ is planned and substituted into blind-hole theory with the application of the least squares method and within the recommended range of the parameters. It can be observed that its error of calculation results is lower than $\bar{a}$ , $\bar{b}$ .

Footnotes

Handling Editor: Stephen D Prior

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Chao-Ming Hsu

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