Proceedings of the SEM Annual Conference June 1-4, 2009 Albuquerque New Mexico USA ©2009 Society for Experimental Mechanics Inc.

Numerical Study on Calibration Coefficients for Hole-drilling Residual Stress Measurement

Bowang Xiao1, Keyu Li2 and Yiming Rong3 1. PhD candidate in Manufacturing Engineering Department, Worcester Polytechnic Institute, 100 Institute Rd, Worcester, MA 01609, USA, former graduate student in Mechanical Engineering Department at Oakland University, [email protected] 2. Mechanical Engineering Department, Oakland University, MI 3. Manufacturing Engineering Department, Worcester Polytechnic Institute, MA

ABSTRACT Calibration coefficients are used to calculate residual stresses from relieved strains during hole-drilling. Some nondimensional coefficients are published and used to interpolate calibration coefficients for a given measurement. This approach is simple to avoid complicated coefficient calibration through performing experiments and finite element simulations, but it introduces two problems. Firstly, errors are always introduced from interpolation. Secondly, The FE model used for the published coefficients is a general model which cannot reflect special conditions e.g. the varied model dimensions. How the calibration coefficients vary with respect to various factors is numerically studied in this paper. The various factors studied include sample geometry dimensions, the radius of the drilled hole, offset and incline and material properties. Based on the study, a set of routines coded in Python language for Finite Element software ABAQUS is developed to obtain the calibration coefficients for a specific measurement condition. A demonstration of obtaining the calibration coefficients with these automatic routines is also presented. KEYWORDS Calibration Coefficients, Finite Element, Incremental Hole-drilling Method, Residual Stress 1. INTRODUCTION In the measurement of residual stress using center hole-drilling method, resistance strain rosette is used to measure relieved strains during the hole-drilling. The measured strains are then used to back-calculate residual stresses, and the integral hole-drilling method was developed to calculate residual stresses in non-uniform residual stress fields [1-2]. The calibration coefficients in the relationship of the residual stress and relieved strains are critical and need to be pre-determined. No exact solution for a blind hole into a field of plane stress is yet available from the theory of elasticity, although there was a closed form solution of the residual stresses for cutting a through-hole on a thin plate [3-5]. The calibration coefficients for blind-hole method can be determined experimentally [3] or by numerical simulation with FE model [1-2, 6-7]. Some dimensionless calibration coefficient tables were obtained by using a 2-Dimenional Finite Element (2-D FEA) model and it was reported that the calibration coefficients for any specific testing case can be interpolated from the dimensionless calibration coefficient data tables [1-2]. However, based on our finding, errors are introduced numerically when interpolating and extrapolating these nondimensional coefficient tables. Because the published coefficient tables are calculated from a general measurement model and a specific measurement may have a different model, the application of coefficients from these general tables to a specific measurement will introduce further errors. How the calibration coefficients vary with respect to various factors such as sample geometry dimensions, the radius of the drilled hole, offset and incline of holes and material properties is numerically studied in this paper. Based on this study, it is recommended to determine these coefficients using FE simulation for a specific measurement condition to eliminate the interpolation errors. In the FEA, automatic routines are developed. An automatic process to calculate these calibration coefficients was proposed to study the influence of gage location and hole radius [8]. But the automatic process is not applicable. A new set of routines programmed in Python language for FE

software ABAQUS is developed in this paper, with which we can get the calibration easily, quickly and conveniently, even for a technician who is not familiar with Finite Element Analysis. A simple demonstration shows that with these routines, it is quite easy and fast to obtain accurate calibration coefficients. 2. PRINCIPLE A strain gauge rosette is mounted onto the surface of the sample and a small hole is drilled at the center of the 3 strain gauges. During the hole drilling, strains are relived due to the release of residual stresses. The relived strains are measured and used to back-calculated residual stresses in that small area. However, no exact solution of residual stresses in terms of relived strains for a blind hole drilled into a plane stress field is available from the theory of elasticity [3, 4]. Fortunately, it had been demonstrated that this case closely parallels the through-hole condition in the general nature of the stress distribution [3, 9]. The general expression for the relieved radial strains due to a plane biaxial residual stress state is Equation 1 [3]:

 1  A( x   y )  B( x   y ) cos 2  2  A( x   y )  B( x   y ) cos 2  3  A( x   y )  B( x   y ) cos 2 where

(1)

 1, 2,3 =measured strain relieved from strain gauge 1, 2 and3, respectively  x ,  y =stress in x and y direction, respectively A, B

 

=Calibration Coefficients =angle measured counterclockwise from the x direction to the axis of the strain gauge 1, 2 and 3,

respectively For a rectangular strain gauge rosette like, the three strain gages measure the three strains along the 3 gage directions during hole-drilling, where  solved and shown in Equation 2:

1   3

 0 o ,   45o ,   90 o . The principal stresses and their direction are

1 ( 3   1 ) 2  ( 1   3  2 2 ) 2 4A 4B   1  min  1 3  ( 3   1 ) 2  ( 1   3  2 2 ) 2 4A 4B   2 2   3 tan 2  1 1   3

 max 



(2)

Since the coefficients A and B for blind hole-drilling cannot be calculated directly from theoretical considerations, they are usually obtained by numerical procedures such as finite-element analysis [3]. Some tables of the coefficients defined in Equation 3 were published [2]. It is suggested that the coefficients A and B can be interpolated or extrapolated from the published nondimensionless coefficients [1-2]. However, errors are always introduced in this procedure. More accurate residual stresses can be calculated if the errors of interpolation can be avoided, e.g. determine the calibration coefficients directly from experiments or FEA for a specific measurement.

a

2E A 1 

b  2E  B

(3)

3. STUDIES OF COEFFICIENTS SENSITIVITY TO VARIANT FACTORS VIA A 3-D FE MODEL A series of FE simulations were performed to study the effects of element size, material properties, drilled hole radius and inclination angle and sample geometry dimensions.

ELEMENT SIZE -- The sensitivity of calibration coefficients to FE meshes is studied with three difference mesh sizes. For mesh 1 as shown in Figure 1, the sizes of elements in the hole area and gage grid area are 0.3mm and 0.5mm, respectively. Element length in the axial direction for the top portion is 0.1285mm, which gives the ratio of layer depth to mean gage radius as 0.1285mm/2.57mm=0.05. All other elements are in 1.0 mm size. For mesh 2 as shown in Figure 2, these parameters are 0.15mm, 0.3mm, 0.1285mm and 0.5 mm respectively. For mesh 3 as shown in Figure 3, these parameters are 0.08mm, 0.15mm, 0.1285mm and 0.3 mm respectively.

Figure-1.

Position of strain gage rosette in the 3D model mesh 1

Figure-2.

Position of strain gage rosette in the 3D model mesh 2

Figure-3.

Position of strain gage rosette in the 3D model mesh 3

Figure-4.

Boundary conditions and loads applied

The coefficients at the depth of

Z / rm =0.1 and 0.3 are calculated using these three mesh schemes, and

tabulated in Table 1. Young’s modulus and poison’s ratio are chosen as 100GPa and 0.3, respectively. The hole radius is set as 0.771 to make the ratio of hole radius to mean gage radius as 0.3. The coefficients aij and bij are special for integral method, which was proposed for non-uniform residual stress where the hole is drilled layer by layer [1-2]. The coefficient aij is the matrix element in the ith row and jth column for integral method. So is bij . It is seen that the coefficients are converging to constants as element sizes are being halved, but at the same time the CPU time increases dramatically as shown in Table 2. Since the percent differences between results from mesh 2 and mesh 3 are less than 2%, the mesh 2 is chosen with a balance consideration of CPU time.

Table 1.

a11 b11 a33 b33

Calibration coefficients at depths

Z / rm =0.1 and 0.3, respectively

Mesh 1

Mesh 2

Mesh 3

% difference 12

% difference 23

-0.0365 -0.0749 -0.1126 -0.2454

-0.0353 -0.0772 -0.1059 -0.2585

-0.0352 -0.0785 -0.1041 -0.2602

3.50 2.92 6.28 5.05

0.12 1.75 1.61 0.73

Table 2.

Mesh sizes and CPU time Mesh 1 Mesh 2 Node # 9468 44640 Element # 8192 41001 CPU time (s) 13.8 215.6

Mesh 3 90924 84897 1763.2

MATERIAL PROPERTIES -- Former researchers have suggested that the nondimensional calibration coefficients are not sensitive to material properties [2-3]. However, according to our findings, the two nondimensional calibration coefficients are dependent on Poisson’s ratio to some degree. Three groups of calibration coefficients at the depth of Z/Rm 0.3 are obtained using different Poisson’s ratio, 0.25, 0.3 and 0.35, while the Young’s Modulus is set as 100 GPa. The radius of the hole is set as 0.771 to make the ratio of hole radius to mean gage radius as 0.3. The comparisons of the nondimensional coefficients indicate that the calibration coefficients are non-sensitive to Young’s modulus, but, are sensitive to Poisson’s ratio. The coefficient aij changes greater than 5% when Poisson’s ratio changes in the range of 0.25 to 0.35, as shown in Table 3. Table 3. Calibration coefficients vs. Poisson’s ratio Poison’s ratio 0.25 Max. %difference 0.30 0.35

a33

-0.1075

-0.1059

-0.1020

5.06

b33

-0.2561

-0.2585

-0.2585

0.92

DRILLED HOLE DIAMETER -- In the 3D FE model, a uniform stress field (  x =Constant 1, applied to obtain

 y =Constant

2) is

aij and bij in one calculation, as shown in Figure 4. The radius of hole drilled may vary for

different sample materials even if the same drill tool is used. As shown in Table 4, a hole radius change of 0.1 mm may introduce an uncertainty up to 27%, and similar conclusion can also be reached from the published coefficient tables [1-2]. Therefore, for every measurement it is necessary to measure the hole radius with highsensitivity optical facilities and determine the calibration coefficients corresponding to the hole radius. Table 4.

Calibration coefficients vs. hole radius

ra (mm)

0.75

0.8

0.85

Max. %difference

a33

-0.1011

-0.1140

-0.1259

24.57

b33

-0.2465

-0.2784

-0.3125

26.76

DRILLED HOLE OFFSET -- In practice, it is difficult to drill the hole at exact the center of the strain gage circle. The calibration coefficients sensitivities to hole center are studies by applying offsets of  0.1mm and  0.2mm. A half-model as shown in Figure 5, other than a quarter-model, is used to obtain the calibration coefficients when hole offset exists. The calibration coefficients with negative offset are calculated from strain gage 1’ and strain gage 3, while the ones with positive offset are calculated from strain gage 1 and 3. The results show that a hole offset of 0.1 mm may introduce an uncertainty of less than 3.9%, as shown in Table 5. In reality, the offset can be

controlled smaller than 0.1 mm if hole drilling procedure follows the ASTM standard [5] with professional holedrilling equipment. DRILLED HOLE INCLINE -- The hole incline, shown in Figure 6, exists when the surface is not flat, or when the drilling equipment is not so accurate. The hole incline may also be introduced when we cannot drill the hole perpendicularly to the surface due to complex geometry of the sample. The calibration coefficient stability when o o o the hole incline angles are 2.5 , 5 and 10 is studied and results are tabulated in Table 6. The results show that aij and bij may vary less than 2.2% if the hole incline angle can be controlled less than 5 o.

Gage 3

RSR Gage 1’

Gage 1

Center of strain gage Figure-5. Table 5. offset

Offset

Incline angle

Center of hole drilled

Hole offset scheme layout (top view)

Calibration coefficients vs. hole offset %difference 0 0.1mm 0.2mm

Figure-6.

Hole drilled

hole incline (side view)

%difference

a33

-0.1140

-0.1175

3.06

-0.1194

4.710

b33 offset

-0.2784 0

-0.2675 -0.1mm

3.89 %difference

-0.2508 -0.2mm

9.904 %difference

a33 '

-0.1140

-0.1139

0.06

-0.1106

3.039

b33 '

-0.2784

-0.2814

1.10

-0.2853

2.475

Table 6. Calibration coefficients vs. hole incline o % difference Incline Angle ( ) 0 2.5 5

10

%difference

a33

-0.1169

-0.1161

-0.1152

1.438

-0.1144

2.10

b33

-0.2608

-0.2632

-0.2664

2.150

-0.2753

5.56

MODEL DIMENSIONS -- An assumption of infinite big sample was made in order to calibrate the coefficients by either experiments [3] or FEM [1, 2, 6 - 8]. Some studies were conducted on the calibration coefficient sensitivity to the thickness of the model while assuming the model is an infinite plate and it was found out that the thickness affects the coefficients significantly especially when the thickness is less than 1.5 times of mean radius of rosette ( rm ) [6-7]. In reality, the measurements often took place on a narrow surface, where the coefficients must be calibrated according to its real dimension. As shown in Table 7, the sensitivity study in this paper shows the influence of plate length and width, which is similar to that of plate thickness. Table 7. Calibration coefficients vs. part width half width(mm) 6 8 10 12.5

a33

-0.1105

-0.1131

-0.1138

-0.1142

Max. %difference 3.2438

b33

-0.2623

-0.2730

-0.2764

-0.2779

5.6173

CONCLUSIONS OF SENSITIVITY STUDIES -- The coefficients are sensitive to element size, sample geometry dimensions, radius, offset and incline of the drilled hole, and material properties. Therefore, it is necessary to calculate the calibration coefficients for the specific measurement conditions every time. Automatic routines coded in Python language for FE package ABAQUS to calculate calibration coefficients for specific measurement conditions eliminate the interpolation errors and save the troubles of running FE simulation manually, which is presented in section 4. 4. AUTOMATIC ROUTINES FOR ABAQUS ABAQUS is a suite of powerful finite element package, which can solve problems ranging from relatively simple linear analyses to the most challenging nonlinear simulations. The ABAQUS Scripting Interface is an application programming interface (API) to the models and data used by ABAQUS and is an extension of the Python objectoriented programming language. It can create and modify the components of ABAQUS models, submit ABAQUS analysis jobs and read ABAQUS output databases. [10-11]

Technician

Relieved strains Measurement conditions

Python program

ABAQUS package

Center hole-drilling measurement [3]

 Calibration coefficients Output automatically  Residual stresses Figure-7.

Automatic determination of calibration coefficients and residual stresses

A series of routines were coded in Python so that the coefficients corresponding to sample geometry dimensions, materials, drilled hole radius, load magnitude and mesh schemes are obtained and residual stresses are backcalculated with these coefficients. As shown in Figure 7, a technician conducts the residual stress measurement and then inputs the relieved strains and measurement conditions to the Python routines, which will call ABAQUS to generate 3-D FE model based on user specified thickness, width and length, material properties, depth of cutting hole, loads, boundary conditions, etc. and run simulations automatically. The simulation results are fetched by Python code so that coefficients can be determined automatically. Therefore, residual stresses can be

calculated based on relieved strains and the automatic determined calibration coefficients by the Python code. In this procedure, the technician conducts the measurement as usual and does not necessarily know how to run FE software. The troubles of interpolating calibration coefficients are also saved. Here an application of these automatic routines was demonstrated with a case of measuring residual stresses on aluminum beam which is 3.18 mm thick and 19.07 mm wide. A hole-drilling test is taken as an example with increments of 0.127mm, 0.127mm, 0.254 mm, and 0.254 mm. First, parameters for dimension, radius (0.8mm) and depth of the hole, material properties (E=72GPa, v=0.32), were assigned values corresponding to this measurement. Then the Python routine calls ABAQUS to create an FE model as shown in Figure 8 and runs the simulation. And then the coefficients for this measurement are determined by the routines, which are tabulated in Table 8 and 9. In this procedure, the assignment of the measurement conditions is the only thing in which the technician is involved. From this demonstration, it is seen that more accurate calibration coefficients corresponding to real measurement conditions can be determined very easily and fast with the automatic routines. Thus that, more accurate residual stresses can be calculated from relived strain using these accurate calibration coefficients.

Figure-8. Table 8.

Coefficient

Automatic generated mesh scheme for coefficients

Aij (units in  )

Table 9.

Coefficient

MPa

h

H 0.05

0.1

0.2

0.05 0.1 0.2 0.3

-0.1714 -0.2303 -0.2972 -0.3349

-0.1721 -0.2512 -0.2811

-0.3221 -0.4075

0.3

-0.1873

Bij (units in  ) MPa

h 0.05 0.1 0.2 0.3

H 0.05 -0.2971 -0.3736 -0.4597 -0.5097

0.1

0.2

0.3

-0.3166 -0.4351 -0.4814

-0.6464 -0.7787

-0.4725

5. CONCLUSIONS Interpolating calibration coefficients from the published tabulated dimensionless coefficients produces considerable errors. It is also found that the coefficients are sensitive to element size, sample geometry dimensions, radius, offset and incline of the drilled hole, and material properties. In this paper, a set of routines coded in Python were developed to determine the coefficients corresponding to a specific measurement to improve the accuracy and convenience. 6. ACKNOWLEDGEMENT This work was supported financially by General Motor Company, Michigan, USA. Dr. Qigui Wang from GM Powertrain is much appreciated for his kind help and support. REFERENCES: 1. Schajer, G. S., ‘Measurement of Non-Uniform Residual Stresses Using the Hole Drilling Method, Part I— Stress Calculation Procedures’, ASME Journal of Engineering Materials and Technology, 110, No. 4, pp. 318–342, 1988

2. Schajer, G. S., ‘Measurement of Non-Uniform Residual Stresses Using the Hole Drilling Method, Part II— Practical Application of the Integral Method’, ASME Journal of Engineering Materials and Technology, 110, No. 4, pp. 344–349, 1988 3. Vishay company technology, ‘Measurement of Residual Stresses by the Hole-Drilling Strain Gage Method’, Strain Measurement Technology, http://www.vishay.com/brands/measurements_group/guide/guide.htm, 1997 4. Timoshenko, S. and J.M. Goodier, Theory of Elasticity, New York: McGraw-Hill, 1951 5. ASTM E 837, ‘Standard Test Method for Determining Residual Stresses by the Hole-Drilling Strain-Gage Method’, American Society for Testing and Materials, Philadelphia, Pa, USA, 2002 6. Jong-Ning Aoh, Chung-Sheng Wei, ‘On the Improvement of Calibration Coefficients for Hole-Drilling Integral Method: Part 1—Analysis of Calibration Coefficients Obtained by a 3-D FEM Model’, ASME Journal of Engineering Materials and Technology, Vol. 124, 2002 7. Jong-Ning Aoh, Chung-Sheng Wei, ‘On the Improvement of Calibration Coefficients for Hole-Drilling Integral Method: Part 2— Experimental Validation of Calibration Coefficients’, Journal of Engineering Materials and Technology, Vol. 125, 2003 8. Sicot, O., Gong, X.L., Cherouat, A. and Lu, J., ‘Influence of experimental parameters on determination of residual stress using the incremental hole-drilling method’, Composites Science and Technology 64, pp.171180, 2004 9. Rendler, N.J. and Vigness, I., ‘Hole-drilling Strain-gage Method of Measuring Residual Stresses’, Proc., SESA XXIII, No. 2: 577-586, 1966 10. ABAQUS, ABAQUS help documentation, Hibbitt, Karlsson & Sorenson, Inc, Pawtucket, RI, USA, version 6.5 11. Python Software Foundation, Python documentation, http://www.python.org/ , version 2.5