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Strain sensitivity enhancement for the hole-drilling residual stresses measurement method Tootoonian, Mohammad 1993

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STRAIN SENSITIVITY ENHANCEMENTFOR THEHOLE-DRILLING RESIDUAL STRESSES MEASUREMENT METHODbyMORAMMAD TOOTOONIANB.Sc., Tehran University, Iran, 1980A TILESIS SUBMITTED iN PARTIAL FULFILLMENT OFTILE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIES(Department of Mechanical Engineering)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIADecember 1993© Mohammad TootoonianIn presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of ft( €Cit VCt /The University of British ColumbiaVancouver, CanadaDate A’yte 3e/ 3DE-6 (2/88)ABSTRACTTwo methods for enhancing the strain sensitivity of the hole-drilling method formeasuring residual stress fields were examined in this thesis. Such enhanced strainsensitivity is important because it improves the accuracy of the residual stress evaluation.The first method involves enlarging the effective hole size by drilling a reverse taper hole.A simple practical technique for drilling reverse taper holes is described. The strainsensitivity for this new method is compared with that of the conventional hole-drillingmethod. Experimental results show excellent correspondence with theoretical results.The reasons for the sensitivity improvement are explained. The second method involvesdesigning a 6-element strain gauge rosette. It is shown that the new 6-element rosettesignificantly enhances the strain sensitivity of the hole-drilling method. Experimentalresults show excellent agreement with predicted results. Moreover, it is shown that thisnew rosette improves the accuracy of the method concerning the measurement of thevariation of residual stresses with depth.11TABLE OF CONTENTSABSTRACT iiLIST OF FIGURES vNOMENCLATURE vilACKNOWLEDGMENT ixCHAPTER 1INTRODUCTION 11.1 Residual Stresses 11.2 Methods ofMeasurement ofResidual Stresses 21.3 The hole drilling method 41.4 Objective and overview of the study 6CHAPTER 2THE HOLE DRILLING METHOD 92.1 Background Theory 92.1.1 Uniform Residual Stress 102.1.2 Non-Uniform Residual Stress 162.2 Strain Sensitivity of the Hole-drilling Method 19CHAPTER 3TAPER HOLE DRILLING 223.1 Drilling a Taper Hole 233.2 Taper Hole Calibration Coefficients 273.3 Experimental Verification 30CHAPTER 4NEW STRAiN GAUGE GEOMETRY 334.1 Influence of strain gauge geometry on the calibration coefficients 334.2 Evaluation of calibration coefficients for different strain gaugegeometries 364.3 Calibration coefficients for different strain gauge geometries 394.4 Experimental verification 47liiCHAPTER 5CONCLUSION.50REFERENCES 53APPENDIX 55Calibration coefficients for a circumferential strain gauge 55ivLIST OF FIGURESFigure 1 Typical strain gauge rosette for the hole-drilling method 5Figure 2 A typical hole geometry 9Figure 3 Stress distribution before and after the hole drilling 13Figure 4 Calibration coefficients for ra /rm = 0.5 15Figure 5 Stress loading corresponding to calibration coefficients 18Figure 6 Cross section of a taper hole 22Figure 7 Photograph of the taper hole drilling jig 23Figure 8 Taper hole drilling device 24(a) = cross-section; (b) detail of drill bit; A = high speed turbine;B = adjustment micrometer; C = upper slide tube; D = lower slider;E = ball bearing, F = base; G = inverted cone drill bit; H = specimen;I = diagonal micrometer; A-A = fixed central axis; B-B = drilling axis.Figure 9 Measurement of taper hole diameter by a modified caliper 26Figure 10 A cross-section of a sample drilled taper hole 26Figure 11 Detail of the finite element mesh for a tapered hole 27Figure 12 Theoretical calibration coefficients for straight and taperedholes for c/rm=0.5 28Figure 13 Exaggerated displaced shape profile of straight and tapered holes 29(a) straight hole (b) taper holeFigure 14 Tension test sample 30VFigure 15 Comparison of experimentally measured calibration coefficientswith theoretically predicted values 32Figure 16 Radial strain gauge rosette 35Figure 17 Circumferential strain gauge rosette 35Figure 18 Calibration coefficients ä and b for three different straingauge designs 38Figure 19 Inverse condition numbers for three different strain gauge designs 41Figure 20 Reliability factors for three different strain gauge designs 42Figure 21 A layout of a 6-element combined radial and circumferential rosette 43Figure 22 Calibration coefficient ä and b for four different strain gaugedesigns 45Figure 23 Reliability factors for four different strain gauge designs 46Figure 24 Comparison of experimental and theoretical ä and b values fora 6-element rosette and a rectangular rosette 48viNOMENCLATUREa, b = normalized form of the calibration coefficients A, Ba,, b = calibration coefficients and b corresponding to a hole i increments deepb, = calibration coefficients for j increment within a hole i increments deep[a], [9 = matrix form of the calibration coefficients and b which have and astheir elementsA, B = calibration coefficients for infinitesimal relieved strainsA, = calibration coefficientsC, CBI = condition numbers of matrices [a] and [9F, FB1 = stress evaluation reliability factorsE = Youngs modulus of elasticityh hole depthra = hole radius for straight holes and surface hole radius for tapered holes= inner radius of a strain gauge rosetterm = mean radius of a strain gauge rosette6 = relieved strain measured by the strain gauge= strains measured by the strain gauge 162 = strains measured by the strain gauge 2vii= strains measured by the strain gauge 3= average (over the strain gauge grid) strain due to hydrostatic stress= average (over the strain gauge grid) strain due to shear stress= total measured relieved strain for i increments deep holev = Poisson’s ratio= angular coordinate of the radial mid axis of the strain gaugemeasured counterclockwise from the maximum principal stress= angular coordinate of the radial mid axis of the strain gauge measuredcounterclockwise from the maximum principal stresses at j increment= maximum principal residual stress= minimum principal residual stress= maximum principal residual stress at the increment juj = minimum principal residual stress at the increment jvii’ACKNOWLEDGMENTThis thesis is dedicated to my wife, Mahnaz. Your love, patience and encouragementhave made it possible.I would like to thank my supervisor Dr. Gary Schajer for his invaluable guidance andenormous patience. I am thankful to Mr. Len Drakes, Mr. Tony Besic, and Mr. AntonSchreinders who professionally manufactured the necessary equipment and samples.This study was financially supported by the Natural Science and EngineeringResearch Council of Canada (NSERC).ixCHAPTER 1INTRODUCTION1.1 Residual StressesResidual stresses are stresses that exist in a material in the absence of any externalloads. They can be induced in almost every step of most manufacturing processes.Usually, a permanent dimensional change in a portion of a work-piece is the source of theresidual stresses. This dimensional change can occur by a plastic deformation, forexample, during rolling, forming, and machining. Plastic deformation also can be causedby the large temperature gradients that occur during welding, heat treating or sintering.Elastic deformation also can be a source of dimensional changes and residual stresses, forexample by tight fitting of assembled components or by solid-state phase transformations.Residual stresses can significantly affect the serviceability of engineering components.The influence of residual stresses on fatigue strength is well known. Fracture, surfacecorrosion, and crack propagation are also directly influenced by to distribution of residualstresses in materials. Residual stresses are one of the important causes of failure ofmaterials. Residual stresses can be beneficial as well as detrimental. Compressive residualstresses are mostly considered beneficial and tensile stresses are generally detrimental.Dimensional stability of a piece is impaired by residual stresses, compressive or tensile.Despite their significance, residual stresses are often ignored in design andmanufacturing, mostly because they are difficult to evaluate. Ever-growing needs forenhanced reliable design of lighter and smaller but safer components demand a betterunderstanding of residual stresses. Reliable methods for measurement of residual stressesare essential for studying these stresses.IConventional methods for measuring stresses due to applied loads in experimentalstress analysis may not be suitable for measuring residual stresses. In conventionalmethods relative stresses are measured, i.e., the applied stresses are measured bycomparing the current state of stress or strain with the zero state of stress or strain (byremoving or applying external loads). However, there are no external loads in the case ofresidual stresses. Thus, specialized stress measurement methods must be used. Thesemethods are divided into two categories: non-destructive methods which measure theabsolute state of stress, and destructive methods which involve removing some stressedmaterials [1].1.2 Methods of Measurement of Residual StressesThe non-destructive measurement methods generally measure the properties ofmaterial that are altered by the absolute state of the stress of the specimen. Often therelation between these properties and stresses are not filly established and the results ofthe measurements by non-destructive methods are subject to interpretation. The mostcommonly used of these methods is the X-ray method. Others include the ultrasonic,magneto-acoustic, photoelastic, and the neutron diffraction methods. Following are briefdescriptions of the most widely used of these methods [1].The X-ray method is by far the most developed and the most widely used of the nondestructive methods for measuring residual stresses. It uses X-ray diffraction to measurethe distance between two crystallographic lattice layers. This distance is changed by thestate of stress of the material. Despite its widespread use, the X-ray diffraction methodhas some limitations. For example, the small penetration range of X-rays only permitsmeasuring the surface residual stresses. The neutron diffraction method that is similar to2the X-ray method, has its penetration range of several orders of magnitude larger than theX-ray method, but has a low precision due to weak sources of neutrons.The ultrasonic method uses changes in the velocities of ultrasonic waves due tostresses as a basis to measure residual stresses. One of the shortcomings of this method isthat the average stress is measured, and therefore sharp stress gradients cannot bemeasured.A few well-known destructive methods for residual stress measurement are thedissection method, the ring core method, and the hole drilling method. These methodsevaluate residual stresses by measuring strains or displacements caused by removal ofstressed material.The most powerfbl, and most destructive, method is the dissection method. In thismethod, the stressed material is dissected layer by layer and at each step the deformationof the remaining material due to the removal of the material is measured. The originalresidual stresses can then be calculated. The method is very time consuming and thesample is completely destroyed.Unlike the dissection method, which involves complete destruction of themeasurement area, the ring core and hole drilling methods cause much less damage. Forthat reason they are often referred to as “semi-destructive “ methods. The ring coremethod involves cutting a small ring core in the stressed material and measuring therelieved strains on the surface of the material remaining in the ring. The original existingresidual stresses then can be calculated from the measured relieved strain data. The holedrilling method is similar to core ring method in principle, however, instead of a ring corea small hole is drilled and the relieved strains around the hole are measured. The ring coremethod has a higher sensitivity than the hole drilling method. However the size of theannular ring is relatively large, causing more damage than the hole drilling method.3Moreover, the results are much less localized. In contrast the hole drilling method doesrelatively little damage to the specimen and it is capable of more localized residual stressmeasurement. In addition drilling a hole is easier than cutting a ring core.Despite its relatively low sensitivity, the hole drilling method is the most widely usedtechnique for measuring residual stresses. The popularity of the method is mostly due toits simplicity, reliability and ease ofuse.1.3 The hole drilling methodThe hole drilling method involves drilling a cylindrical hole at the point of interest in astressed material, measuring the relieved strain (or displacement) around the hole andcalculating the original stresses from these measured strains [2]. The hole drilling methodwas first introduced by Mathar in 1934 [31. He used a mechanical extensometer tomeasure the displacements around a through hole in a stressed plate. For his calculationhe used the well-known Kirsch [4] solution for the stress distribution around a small holein a thin plate subject to uniform stress. The accuracy of the method used by Mathar waslow because of the use of a mechanical extensometer. Soete and Vancrombrugge, in1950, greatly improved the accuracy of the method by using electrical strain gauges, inplace of the extensometer [51.In 1956, Kelsey [6] published his investigation into using the hole drilling method tomeasure stress variation with depth. He was also the first to use blind holes instead ofthrough holes. Previously, the use of through holes limited the method only to thin plates.The method gained became standardized in 1966, when Rendler and Vigness developedthe method into a systematic and reproducible procedure [7]. Their work is used as a base4for the ASTM E837 [8] standard for hole drilling method. Figure 1a typical hole-driffing rosette.schematically showsFigure 1 Typical strain gauge rosette for the hole-drilling method.In 1974, Beaney and Procter [9] improved the practical aspect of the method by usingair abrasive machining to enable stress free hole drilling. Flaman [10] developed the useof ultra high speed drilling to achieve stress free hole.Schajer [11], in 1981, provided the first comprehensive finite element analysis of themethod. Later in 1988, he published his systematic investigation on using the hole drillingmethod for determination of residual stress variation with depth and provided the finiteelement analysis for the case [12].The technical literature on the hole drilling method is extensive and continues togrow.51.4 Objective and overview of the studyDespite its widespread use and popularity, however, a weakness of the hole drillingmethod is that the measured strain reliefs are quite modest in size. The inevitable smallexperimental errors that occur may not be trivial compared with the measured strains. Insuch cases, the small errors could significantly impair the accuracy of the computedresidual stresses.In most hole-drilling measurements, it is assumed that the stresses in the specimenmaterial do not vary with depth from the measured surface. However, in recent years,considerable interest has arisen for using the hole-drilling method to determine residualstresses that vary with depth from the measured surface. The associated stresscalculations are numerically very sensitive, and even quite small strain measurement errorscan have devastating effects on the accuracy of the results. Thus, in the non-uniformstress case, it is especially important to make the strain sensitivity of the hole-drillingmethod as high as possible, and to minimize the relative size of the experimental errors.Three key factors controlling the strain sensitivity of the hole-drilling method are thediameter and geometry of the hole and the geometry of the strain gauge rosette. Thestrain sensitivity of the hole-drilling method depends directly on the size of the holerelative to the rosette size. Maximum sensitivity for a given rosette size is achieved whenthe hole has the maximum allowable size. This maximum size is determined by thedistance between the edge of the hole and the strain gauge grids. Unfortunately, evenwith maximum size hole, the sensitivity of the hole-drilling method is not very high. Thismodest sensitivity means that small strain measurement errors can cause significant errorsin the calculated residual stresses. Two methods are proposed here to improve strainsensitivity and stress calculation accuracy: taper hole drilling and a modified design for therosette.6Taper hole drilling improves the sensitivity of the hole drilling method by increasingthe hole size without exceeding the limit for hole radius at the surface of the specimen. Inthis method, a truncated cone shape hole is drilled instead of a conventional cylindricalhole. By modifying the geometry of the hole, the effective size of the hole and theflexibility of the material in the region close to the hole are increased. This increase inflexibility of the material causes larger strain reliefs and improved sensitivity. This thesisdescribes and explains the sensitivity improvement that is achieved by tapered hole drilling.A practical drilling procedure is also briefly described. Experimental measurement of thesensitivity is compared with numerical results.The geometrical design of the strain gauge rosette is the third key factor whichinfluences the sensitivity of the method and its stress calculation accuracy. The ASTMstandard hole-drilling rosette [8] is the most commonly used design, and is almostuniversal in North America. The pattern derives from the original 1966 work of Rendlerand Vigness [7]. These two pioneering researchers do not give much detail concerningtheir choice of rosette geometry. One can speculate that theirs was a pragmatic choicebetween strain sensitivity and available strain gauge shapes. Certainly, their final selectionhas served well over many years. However, the more recent requirements for improvedstrain measurement accuracy for non-uniform stress evaluations heavily tax the capabilitiesof their rosette design.This thesis examines how various rosette design factors contribute to overall strainsensitivity. It compares four different potential rosette geometries, and suggests animproved design that has an effective strain sensitivity almost three times greater than thepresent the ASTM standard pattern. The non-uniform stress calculations associated withthe proposed rosette patterns are also numerically less sensitive than those for the ASTMdesign. Non-uniform residual stress profiles can be determined to depths about 25%greater than previously. Furthermore, the proposed pattern includes thermal strain7compensation, so the absolute sizes of the strain measurement errors are also reduced. Aseries of experiments was undertaken using the proposed strain gauge pattern, and thevarious features of the design were successfully demonstrated.8CHAPTER 2THE HOLE DRILLING METHOD2.1 Background TheoryThe hole drilling method involves drilling a small hole into the stressed material. Aspecially designed strain gauge rosette measures the associated strain reliefs in thesurrounding material [7,8]. Figure 2 shows a typical hole geometry. The residual stressesoriginally existing at the hole location can then be evaluated from the measured strains. Inmost hole-drilling measurements, it is assumed that the stresses in the specimen materialare uniform with depth from the surface. However, in recent years, considerable interesthas arisen for using the hole-drilling method to determine residual stresses that vary withdepth from the specimen surface. Both cases are discussed in this thesis.Figure 2 A typical hole geometry.stress afterdrilling92.1.1 Uniform Residual StressWhen the residual stresses in a specimen do not vary with depth from the surface, therelationship between the measured relieved strains after the hole drilling and originalexisting residual stresses is of the form [7]:6 = A(Jmax+Eymin) +(Uma—c7)coS2cP (1)where6 = relieved strain measured by the strain gaugeumax = maximum principal residual stressminimum principal residual stress(0 = angular coordinate of the radial mid axis of the strain gaugemeasured counterclockwise from the maximum principal stress,A, = calibration coefficientsThe principal stresses and their orientations can be calculated by applying the aboveEquation for each of the strain gauges.—8 + 63 ((262 6 — 83)2 + (8 — 83)2)1/2 2Umax0min—()1 282—61—83(0 = —arctan (3)2 61S310where8i, 82, and 63 = strains measured by the three strain gauges= clockwise angle from gauge 1 to direction ofNumerical values of the calibration coefficients A and must be known to determineresidual stresses using Equation (2). For the idealized case of infinitesimal relieved radialstrains around a through hole in a thin uniformly stressed plate the coefficients A andcan be determined analytically at a distance r from the hole radius as [2]:A-B = — 1 + v F 4 (2 (-41 (5)2E Ll+vr) isrJjwhere ra is the hole radius. In Equations (4) and (5), A and B are written without “bars”to emphasize that they refer to infinitesimal relieved strains. The correspondinginfinitesimal A and B values for circumferential relieved strains are:A = 1-v (6)B = 1+ F 4 v fl2 - (7)2E [1+vr) rJ]In the early years of the hole-drilling method, constants A and B from Equations (4)and (5) were sometimes used for hole-drilling residual stress calculations. However, thispractice gives inferior results because practical relieved strain measurements occur overthe entire areas of the strain gauges, not just at infinitesimally small areas. The values of11the calibration coefficients A and can be calculated by integrating the values of A and Bover the active areas of the strain gauges, [11].In most cases, through holes are not practicable and blind holes must be used instead.When a blind hole is used, there are no convenient analytical values for the calibrationcoefficients A and . For this case, values of A and are obtained either experimentallyor numerically.In experimental evaluation of the calibration coefficients [7, 8], a separate sample ofthe specimen material is used. A strain gauge rosette identical to actual residual stressmeasurement is installed on the sample surface. A hole with same geometry as the hole inactual residual stress measurement is drilled into this sample. By subjecting the sample toa known uniform stress field and by measuring the strains before and after the hole drilling,the calibration coefficients can be determined. The experimental calibration has theadvantages of conceptual simplicity and accounting for the procedural influences andmaterial-dependent effects on measured strain response. One disadvantage is that aseparate time consuming calibration is needed for every material, hole size, and rosettegeometry.Numerical determination of the calibration coefficients A and was made possible bythe development of the finite element method [11]. Numerical calibration is more generaland covers a wide range of measurement conditions. Numerical calibration closelymatches the experimental calibration, and is widely accepted and used.Figure 3 schematically shows the state of stress in a specimen both before and afterhole-drilling. Figure 3(a) shows the stresses before hole drilling, including the stresses thatexist at the boundary of the hole which is about to be drilled. Figure 3(b) shows thestresses after the hole drilling. The far boundary is assumed to be sufficiently distant thatthe stresses there are unaffected by the hole drilling. Figure 3(c), which shows the stress12difference between Figure 3(a) and Figure 3(b), represent the stress and strainredistribution caused by hole drilling. An analysis of the state of stress in Figure 3(c)directly gives the strain reliefs measured after hole drilling. The state of stress of Figure3(c) can be modeled by the finite element method to determine the displacement field inthe area surrounding the hole. This displacement field can be numerically integrated overthe active strain gauge area to simulate the measured relieved strains [13]. Substitutingthese calculated strains and known stresses in Equation (1) yield the calibrationcoefficientsA and .HOLE LOCATION/___ ____ ___ ____i!i-tJL_1=L_I(a) (b) (c)Figure 3 Stress distribution before and after the hole drilling.13Numerical calculations for each of the calibration coefficients A and are doneseparately. In this way by applying a hydrostatic stress = = 1 coefficient A canbe evaluated from Equation (1).2 (8)where 8h is average strain (over the strain gauge grid) due to hydrostatic stress. Similarlycan be calculated as(9)2cos2çowhere c is the average strain due to shear stress (om — = 1) and g is the anglebetween the strain gauge mid radial axis and the principal stress direction.The numerical values of the calibration coefficients depend on the hole diameter,hole depth, geometry of the strain gauge rosette, and the specimen material properties,Young modulus B and Poisson’s ratio v. The material property dependency can be mostlyeliminated by introducing two dimensionless calibration coefficients and b [11]:2EA b=2E (10)1+ VIn terms of these two constants, Equations (1) and (2) become6= (1+v)ã (maxmm) +-!? (maxmin) cos2q, (11)E 2 E 21 2 2’f2E (6, +83) — E k(282 — 6 —83) + (8, —83) ,,°max, nn = I \ — + — (12)i1+vj 2a 2b14Further non-dimensionalization is effective in simplifying the hole radius and depthdependencies. Normalizing these two quantities with respect to the mean radius, rm, ofthe strain gauge rosette make the calibration coefficients ä and b (or A and )approximately proportional to the square of the normalized hole radius [11]. Figure 4shows the variation of ä and b with normalized hole depth for a normalized hole radiusof r/rm =0.5 [11]. The corresponding values of A and for a given material can beevaluated through Equation (10). The numerical values of the calibration coefficients,then, can be substituted in Equation (2) to evaluate the residual stresses.0.80.6z• —CC•—-.Li•—0.20h/rmFigure 4 Calibration coefficients for ra /rm = 0.5.0.750 0.25 0.5depth,152.1.2 Non-Uniform Residual StressThe assumption that the residual stresses do not vary with depth is not always valid.In many practical cases the residual stress fields are significantly non-uniform. Processessuch as shot peening, plating, and heavy grinding induce non-uniform stresses.In measurement of the non-uniform residual stress field by the hole-drilling method,relieved strains are measured after drilling of several successive small increments of holedepth [12, 14, 15]. The analysis of the variation of these measured relieved strains withdepth can be used to determine the original non-uniform stress field. There are a fewstress calculation methods. Among these methods the Integral method probably is themost general [12].The basis for the Integral method is that the strains measured during hole drilling arethe cumulative result of relieving the residual stresses originally existing at all depthlocations within the total hole depth. The individual contributions of the stresses at eachdepth location to the total measured strains are identified and the individual stresses arecalculated from these total strain measurements [12, 14].The method involves drilling the hole incrementally. The residual stresses are assumedconstant within each increment. The total measured strain is the sum of the strains causedby relieving the residual stresses originally existing with each of the hole depth increments:6=[(1+1))ii (maxj°mmj) + h mjmuuij) cos2i] (13)where= total measured relieved strain for i increments deep hole160maxj = maximum principal residual stress at the increment jminimum principal residual stress at the increment j= angular coordinate of the radial mid axis of the strain gauge measuredcounterclockwise from the maximum principal stresses at j incrementb = calibration coefficients for j increment within a hole i increments deepThe individual calibration coefficients and b relate the original existing residualstresses and relieved strains.Equation (13) can be written in matrix form as= (1±’’) []{ama)aminj} + []{(maxjminj)c0s2q)} (14)where {6) is a vector of the strain reliefs measured at series of hole depth increments fromthe hole surface. The corresponding stress vectors contain the principal stress quantitiesand directions within each hole depth increment. The calibration coefficients [a} and []become matrix quantities with a lower triangular structure [12]:11 11[a]= a32 a33= : :: (15)Conceptually the stress within each increment can be calculated from the measuredstress {8} by solving the matrix Equation (14). This equation is non-linear, and stresssolutions are usually found in practice using a linear reformulation, [12, 15]. The valuesof the individual coefficients and b depend on the width position of increment j andthe total depth and diameter of the hole. Figure 5 shows a physical interpretation of the17calibration coefficients . The same interpretation applies for b . The columns of thematrices [a] and [b] correspond to the relieved strains caused by the stresses within afixed increment for holes of increasing depth. The increasing hole depth causes theseindividual coefficients to increase within each column. The rows of the matricescorrespond to the relieved strains caused by stresses within successive increments of ahole of fixed depth. The sum of all the coefficients in each row corresponds to a uniformstress field over the entire depth, i.e., and b in Equation (11).a11a21 a22LIa31LIa32LIa33a43Figure 5 Stress loadings corresponding to calibration coefficientsa41 a4218In practice, it is very difficult to produce layered stress fields of the type shown inFigure 5. Thus, the calibration coefficients matrices [a] and [b] are not determinedexperimentally. Finite element calculation, however, provides a very effective and reliablemethod for evaluating these coefficients. A full description of such calculations is given bySchajer [12].2.2 Strain Sensitivity of the Role-drilling MethodThe strain sensitivity of the hole-drilling method is typically quite modest. A lowstrain sensitivity can diminish the reliability of the calculated residual stresses by reducingthe size of the measured strains relative to that of the strain measurement errors. Thestrain sensitivity, defined as the relieved strain per unit residual stress, is quantifiedthrough the calibration coefficients and b. Larger calibration coefficients are desirablebecause they indicate higher strain sensitivity, and hence greater resistance to the effects ofstrain measurement errors.The values of ä and b (or A and B) are low because the strain gauges are located atsome distance from the hole. The material underneath the gauges is subjected only topartial strain relief If it were possible to relieve completely the strains in the materialunder all the gauges, then it may be shown from Hooke’s Law that the correspondingcalibration coefficients would be:maximum value of = 1 V = 0.54 for v = 0.31+ V(16)maximum value of b = 1+ v = 1.3 for v = 0.319Figure 4 shows that for the traditional straight hole, the maximum values of and bare at most only half of the full strain relief values. Thus, significant scope exists forincreasing the strain sensitivity of the strain measurements.Stress calculation accuracy is an additional major concern when using Equation (14)to determine the profile of residual stresses with respect to depth [12, 16]. The matrixquantities [] and {b] become poorly conditioned numerically at larger depths. This effectcauses the calculated stresses at greater depths from the specimen surface to be muchmore prone to error than the stresses at the lesser depth. This error sensitivity can bequantified through condition numbers of the matrices {a] and [b]. A simpleapproximation is given in [121 as:C = and CBI = (17)a11 b1where C and CBL are the condition numbers of matrices [a] and [b] for stresscalculations at increment i. The quantities and b1 are the and b valuescorresponding to a hole i increments deep in a uniform stress field. Their numerical valuesare:(18)The condition numbers C and CB indicate the percent stress calculation errors forthe ith increment caused by one percent strain measurement error within the sameincrement. The larger the condition numbers, the greater is the influence of strainmeasurement errors on the computed residual stresses.Strain sensitivity and matrix numerical conditioning both control how strainmeasurement errors influence the calculated residual stresses. High strain sensitivity and20low condition numbers are desirable to reduce stress calculation error. These two effectscan be combined into stress evaluation reliability factors defined as:F = and FB1 (19)High values of these factors indicate reduced influence of strain measurement errors.Hole size and rosette geometry both strongly influence the reliability factors. Holesize mainly affects the strain sensitivity. Rosette geometry affects both strain sensitivityand condition numbers. The objective of this study is to improve both the strain sensitivityand numerical conditioning of hole drilling method. This objective will be approached intwo ways. The first way will be to modify the shape and size of the drilled hole. Thesecond way will be to modify the geometry of the strain gauge rosette. This study willconcentrate on the effect of these two factors, and how they can be affected to producebeneficial results.21CHAPTER 3TAPER HOLE DRILLINGThe strain sensitivity of the hole-drilling method increases with the increasing size ofthe hole relative to the rosette size. The maximum allowable size is determined by thedistance between the edge of the hole and the strain gauge grids. However, this hole sizelimit is not quite as restrictive as it may appear. The limitation applies strictly only to thatpart of the hole that intersects the specimen surface. By drilling a reverse tapered hole asshown in Figure 6, instead of a straight hole it is possible to increase the effective sizewhile maintaining the hole size limit of the surface. More stressed material is removedclose to the measurement area and more strain is relieved locally.Figure 6 Cross section of a taper hole.relieved stres stress afterdrilling223.1 Drilling a Taper HoleA reverse taper hole can be cut using an inverted cone drill bit in a high-speed air-turbine drive. Figure 7 shows a photograph of the jig that was designed to drill reversetaper holes for this study. Figure 8a shows a schematic cross-section of the drilling jigused in this study. A high-speed air turbine drive “A” mounts inside an upper slide “C”,secured by a vertical height adjustment micrometer “B”. A diagonal motion, at 50° to thehorizontal, is provided by a dovetail slider “D”, and is controlled by a micrometer “I”. Theslider is secured to the working specimen “H” through a ball bearing “E”. The bearing hasa split inner ring to eliminate any free play. An inverted cone drill bit “G”, shown in detailin Figure 8b, mounts at the lower end of the high speed air turbine “A”.Figure 7 Photograph of the taper hole drilling jig.23B A(jL&1fl drill-1 9’ rotation I orbit aroundI hole centerE_ _F/NB A direction of feed motion(a) (b)Figure 8 Taper hole drilling device: (a) = cross-section; (b) = detail of drill bit; A =high speed turbine; B = height adjustment micrometer; C = upper slide tube;D = lower slider; E = ball bearing, F = base; 0= inverted cone drill bit; H =specimen; I diagonal micrometer; A-A fixed central axis; B-B = drillingaxis.24The taper drilling procedure starts by adjusting the diagonal adjustment micrometerso that the air turbine “A” and the drill bit “G” are centered relative to the ball bearing “E”(axis A-A). The vertical height micrometer “B” is then adjusted so that the drill bit “G”just touches the surface of the specimen “H”. Drilling proceeds by incrementally loweringthe diagonal micrometer “I”, causing the drill bit to move diagonally downward, Thediagonal motion displaces the relative axis of the drill bit from A-A to B-B. Rotation ofthe bearing “E” causes the drill bit axis B-B to orbit around the fixed axis A-A, thuscreating a reverse taper hole. This hole can be enlarged by further lowering of the upperslide by the diagonal micrometer “I” and repeated orbiting by rotating the bearing “E”.After drilling, the surface diameter of the taper hole must be measured. Conventionalmeasurement tools are not suited to measure the diameter of small taper holes. Anelectronic vernier caliper was modified by silver soldering two 2.4 mm diameter balls tothe tips of the caliper fingers, as shown in Figure 9. The caliper measures the distancebetween the bails when the balls touch the perimeter and the bottom surface of the taperhole. The surface diameter of a taper hole can then be calculated from the distancebetween the balls and depth of the hole.To confirm the reliability of the modified caliper, eight sample taper holes weredrilled. A caliper measurement was made on each hole, from which the surface diameterwas calculated. Typical surface diameters were in the range 5.8-6.1 mm. Then, surfacediameters were measured in a different way. Each hole specimen was sectioned so that adiameter of the hole was exposed. The surface diameter was then measured opticallyusing a traveling microscope. Figure 10 shows a cross-section of a typical taper hole.The root mean square difference between the surface diameter measurements from thetwo methods was found to be 0.03 mm.25Figure 9 Measurement of taper hole diameter by a modified caliper.Figure 10 A cross-section of a sample drilled taper hole.diameter263.2 Taper Hole Calibration CoefficientsEquations (11) and (1) still apply to taper hole drilling. However, the numericalvalues of the calibration coefficients ä and b (or A and B) are larger than when drilling astraight hole. They can be evaluated, as in the case of a straight hole, either by experimentor by numerical calculation.The numerical procedure for determining and b for a taper hole by finite elementcalculation is similar to the calculation for the case of a straight hole. As shown in [11],the strain reliefs caused by hole drilling correspond to the strains induced by applyingstresses to the hole boundaries which are equal in magnitude but opposite in sign to theexisting residual stresses. The surface displacements caused by this loading are calculatedusing the finite element method. The strains over the strain gauge area can then bedirectly determined from the surface displacements [13].I— hole area ,— specimen surface\\\\\\\\\\\\\\\\c\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\, \ \‘ ‘ \ \\\\\\\\\‘\\\-\ \ \ \ \ \ \\ \ \ \\\\\\\\\ \ \ \\\\\\\\\\\ \ \ \ \\\\\\.c,\ \ \ \\\\ \ -\ \ \ \\‘‘\\\ r\ \ \—zFigure 11 Detail of the finite element mesh for a tapered hole.27A finite element mesh of 432 nodes and 396 elements was used for the calculation.Figure 11 shows the portion of the mesh for the area around the hole. The depth of thehole is adjusted during the various calculations by assigning a near zero elastic modulus tothe elements within the required hole depth. Figure 12 shows a result of thesecalculations, a graph of the calibration coefficients ä and b, for the taper hole case. Forcomparison, Figure 12 also shows calibration coefficients for the straight hole case.As can be seen, the calibration coefficients for taper hole drilling compare favorably intwo different aspects with their corresponding straight hole values; they are larger inmagnitude, and have their peaks at shallower depths. These two features allow greatermeasurement accuracy to be achieved while causing less damage to the specimen.10.8rjO.6O.400 0.25 0.5 0.75Hole depth, h /rmFigure 12 Theoretical calibration coefficients for straight and tapered holesfor ra/rm=O.5.28Figure 13 Exaggerated displaced shape profile of straight and tapered holes.(a) straight hole (b) taper holeFigure 13 explains the reason for these favorable changes. The figure showsexaggerated views of the displaced shapes around a straight hole and a taper holesubjected to a unit biaxial stress field. In the taper hole case, the displacements aregenerally much larger than those of the straight hole case. This feature implies that thetaper hole drilling more effectively relieves the residual stresses around the hole, and thatthe remaining adjacent material is more flexible than in the straight hole case. Thesefactors combine to increase the rate of strain relief during taper hole drilling, i.e., theincrease in initial slopes of the curves in Figure 12. Maximum strain reliefs are alsoreached sooner, so that the peaks of the curves for taper holes in Figure 12 occur atsmaller depths than those for straight holes.—C—C.0.50.51.53 4 5radius I hole radius(a)1 2-0.500.51.54 5radius I hole radius(b)293.3 Experimental VerificationExperiments were conducted to verify the theoretical results in Figure 12. Theexperiments had three objectives: (1) to determine how well the finite elementcalculations can predict the actual ä and b values measured experimentally, (2) tocompare the experimental calibration coefficients for straight hole and taper holes, (3) todemonstrate the practicality of using taper holes for hole-drilling residual stressmeasurements.Two 125-RE strain gauge rosettes were installed on a 12.7 x 38.1 x 565 mmaluminum bar sample shown in Figure 14. Two additional single strain gauges wereinstalled on the sides of the sample to monitor any bending loads. Undesirable bendingstrains were minimized by applying the loads through loading pins inserted through holesdrilled at each end of the sample. The additional strain gauges mounted on the side facesof the test sample confinned that bending strains, both within and out of plane, were lessloading pinstrain gaugerosettestraingaugesFigure 14 Tension test sample.30than 2% of the tensile strain. The sample was subjected to a range of tensile loads fromzero to 35 kN in 7 kN increments, and the corresponding strains were recorded. Themeasured response of the strain gauges per unit load was determined from the gradient ofthe strain versus load plots. This procedure reduces the effect of random strainmeasurement errors and of any existing residual stresses.A taper hole was drilled into the material at the center of the strain gauge rosette infour consecutive 0.58 mm depth increments. Drilling was done by the device described inSection 3.1. After drilling each increment, the sample was subjected to the same range ofloads and the corresponding strains were recorded. Again, the response of the straingauges per unit applied load was determined from the gradient of the strain versus loadplots. After the drilling, the surface diameter of the taper hole was measured as 5.79 mm.This corresponds to a normalized hole radius ra/rm 0.564. From these data, thecalibration coefficients ä and b for a taper hole were evaluated using the methoddescribed by Rendler and Vigness [7]. To provide comparative results, the entireexperimental procedure was repeated using conventional straight hole drilling, using a holediameter of 5.72 mm corresponding to a value of r8 /rm = 0.557.Figure 15 compares the experimentally determined calibration coefficients ä and bwith the theoretical values from the finite element calculations. The results showexcellent agreement between numerically calculated calibration coefficients and theirvalues determined by experiment. The differences between experimental results andtheoretical predictions are typically less than 2%, reaching to 3% only in extreme cases.The experimental results confirm the theoretical prediction that taper hole drillingsignificantly increases the strain sensitivity of the hole-drilling method. Also, strains arerelieved more rapidly so that shallower holes can be used. The experimental work also31C.?CC.?C• —confirms the practicality of drilling reverse taper holes instead of the conventional straightholes using a drilling jig such as the one shown in Figure 8.10.80.60.40.200.5Figure 15 Comparison of experimentally measured calibration coefficients withtheoretically predicted values.0 0.1 0.2 0.3 0.4Hole depth, h /rm32CHAPTER 4NEW STRAIN GAUGE GEOMETRYThe second key factor that affects the strain sensitivity of the hole-drilling method isrosette geometry. Rosette geometry also strongly influences the condition numbers(Section. 2.1.2). This chapter focuses on rosette geometrical design, and identifiesfeatures that lead to improved strain sensitivity and stress evaluation reliability factors.4.1 Influence of strain gauge geometry on the calibration coefficientsThe strain sensitivity of the hole drilling method is directly indicated by thecalibration coefficients and b. However the effects of these calibration coefficients onstress calculation accuracy of the method are not equal. Because the calibrationcoefficient has a smaller value than b, it has a more detrimental influence on the stresscalculation accuracy. This effect can be seen in Equation (12). The first term on the rightside represents the isotropic (equal biaxial) stress component, and the second termrepresents the shear stress component. On average, these stress components have similarmagnitudes. However, the value of ä in the first term is much smaller than that of b inthe second term. This difference makes the first stress term more sensitive to strainmeasurement errors than the second term. The calculated principal stress values combinethe errors in both terms, particularly the larger error. Thus, for most effective stresscalculation accuracy improvement, effort should be concentrated on increasing the valueof ä, preferably so that it reaches the value of b.The length, the difference between inner and outer radii of the strain gauge,significantly influences strain sensitivity. This can be seen from the fact that the strainsaround the hole diminish rapidly with distance from the hole. Therefore, the smaller the33length of the strain gauge a greater fraction of the strain gauge will be in the high-strainregion close to the hole.Shorter strain gauge lengths enhance the numerical values of and significantly.However, a short gauge length can reduce the active area of the gauge and reduce itscapacity as a heat sink. Such a gauge will be more susceptible to thermal drift. The activearea of the strain gauges can be a measure of thermal stability of different strain gaugedesigns. For comparable thermal stability, the different strain gauge designs investigatedin this study have equal active areas.The other important geometrical factor that influences strain sensitivity is theorientation of grid lines of the strain gauges. To study the effects of this factor on thevalues ä and b, two new strain gauge designs with their grid lines oriented radially andcircumferentially were studied. Comparison of these two designs with the existingrectangular design helped to design a more effective pattern for the strain gauge rosette.The active areas of these two new strain gauges were designed to be the same as thestandard ASTM strain gauge. Figure 16 shows the radial strain gauge rosette and Figure17 shows the circumferential rosette.34Figure 16 Radial strain gauge rosette.Figure 17 Circumferential strain gauge rosette.354.2 Evaluation of calibration coefficients for different strain gauge geometriesTo investigate the influence of strain gauge geometrical design on the numericalvalues and conditioning of the calibration coefficients the radial and circumferential straingauge designs shown in Figures 16 and 17 were examined. For comparison a rectangularmodel, which has the same geometry as conventional ASTM strain gauge rosette shown inFigure 1 was also considered. The calibration coefficients, condition numbers andreliability factor for these strain gages for uniform and non-uniform stresses are calculated.The thickness and width of the strain gauge filaments are kept the same as the standardASTM rosette. The active area of the strain gauge grid is the same as the standard rosetteto maintain thermal stability. These strain gauges have the same inner radius as ASTMrosette so that the maximum allowable hole sizes are the same.Calibration coefficients and b were calculated by averaging the finite elementcalculated surface displacement data [13] over the strain gauge grid [11]. Thesedisplacement data are the same for each different strain gauge geometry; however, therequired calculations are different for each different geometry.For the calculation of average strain over the rectangular and radial strain gauge thefollowing Equation, derived by Schajer [13], is used.U2i-U11 (20)iwiwhereU11 and U21 = displacements in the direction of the filament i at its two endsL., w1 = length and width of the filament i36Equation (20) is only valid for straight line grids of uniform width and can be used inevaluation of strains for rectangular and radial strain gauge grids. Corresponding valuesof these strain reliefs can be used to evaluate calibration coefficients and b fromEquations (8), (9) and (10). The results of these calculations for rectangular and radialstrain gauge are shown in Figure 18. Note that in this and subsequent graphs, the holedepth is nonnalized relative to the inner gauge radius, r1, rather than the mean radius, rm.This is done so that the graphs all refer to the same maximum hole size, independent ofchanges in strain gauge length.Calculating and b for a circumferential strain gauge needs a different analyticalapproach that is described in the appendix. The final results are given here.E - (U+2V).= and b = 2 B (21)(1+v) rwhereU1 and V1 radial and circumferential displacements at filament i= grid angle of the strain gauge= mean radius of the filament iFigure 18 also shows graphs of calibration coefficients for a circumferential strain gaugebased on the above equations.37CEeC0.00.40.30.20.1-0.1-0.2-0.3depth, h/r1Figure 18 Calibration coefficients ä and b for three different strain gauge designs..jO/Rectanu1ar•Circumfezentialdepth, h/ri210.80.60.40.20-0.2-0.4••••--.-.—•—.-.Radia1....4.—.— ..--—--Rectangular -—-7;Circumferential384.3 Calibration coefficients for different strain gauge geometriesFigure 18 demonstrates that both calibration coefficients and b are larger for aradial strain gauge than a rectangular one. On the other hand, the calibration coefficientsfor a circumferential strain gauge are smaller in magnitude than for a rectangular one. Inaddition the circumferential coefficients have the opposite sign relative to rectangular andradial strain gauge coefficients.Calibration coefficient ä for a radial strain gauge is increased by about 40 % and b byabout 35 %. Two factors cause these improvements: (1) a larger fraction of the straingauge area is in the high-strain region close to the hole boundary, due to the shorter lengthof the radial strain gauge relative to rectangular design. (2) by using radial filaments, thenegative effect of the circumferential strain, which is present in the rectangular straingauge grid, is eliminated, An additional advantage ofusing a radial strain gauge is that thecalibration coefficient curves for this strain gauge reach their maximum values at ashallower depth compared with a rectangular strain gauge; therefore, a shallower holeneeds to be drilled.In contrast, a circumferential strain gauge provides smaller calibration coefficientsthan a rectangular strain gauge. These reductions in the values of calibration coefficientsare due to the smaller circumferential strains around the hole area compared with theradial strains.These results indicate that a radial strain gauge has higher strain sensitivity and isbetter suited to measuring a uniform stress field than the conventional ASTM strain gauge.However, the influence of these new designs on stress calculations for measuring nonuniform stress fields must also be studied.39The effects of the various strain gauges on calculations of the non-uniform residualstress fields can be examined by considering the condition numbers C and CBI Figure 19shows a graph of the inverse of these condition numbers, i.e., 1/C and 1/Crn versusincreasing hole depth. Small inverse condition numbers indicate worse numericalconditioning and higher error. These graphs show that the effects of the different straingauge geometries are not very significant for calibration coefficient b. However forcalibration coefficient ä the circumferential strain gauge has the best inverse conditionnumber and the radial one has the worst inverse condition number.Figures 18 and 19 show that the radial strain gauge has higher strain sensitivity butpoorer condition number and the circumferential strain gauge has lower strain sensitivitybut better condition numbers. However, both the strain sensitivity and condition numberaffect the stress calculation errors due to strain measurement errors. Therefore, the effectsof these factors must be considered together. The stress evaluation reliability factors Fand FB are defined in Section 2.2 for this purpose. Higher stress evaluation reliabilityfactors indicate more reliable stress evaluations. Figure 20 shows the values of thesereliability factors for the radial, circumferential, and rectangular strain gauge rosettes.Figure 20 shows that the reliability factors for a radial strain gauge has larger valuesthan for a rectangular one up to the depth 0.4r1 for F and up to depth 0.6r1 for FBI.However the rectangular strain gauge is reliable for measuring residual stresses up todepth 0.7r1 [12, 15, 16]. Therefore the radial strain gauge is useful only when residualstresses are within the 0.4i of the surface. In contrast, the circumferential strain gaugehas larger values for F than the rectangular strain gauge for depths larger than 0.4r1However the corresponding values FBI are very low.401zI..2C0• —-.C0rfjI.C—10.80.200.2 0.4 0.6 0.8 1 1.2depth, h / r1Figure 19 Inverse condition numbers for three different strain gauge designs.0.80.60.40.20-0.2Circumferential-—-. - -—- dialRectangular02040608depth, hIr112.D0(00.6E0C00U,I0CCircumfrential.—°— Radial\ Rectangular41a,004-aC,a,‘4-.Da,a,• 0.08I..0t0.060.040.02depth, hIrFigure 20 Reliability factors for three different strain gauge designs.0.080.060.040.020-0.020.12— Circumferential—°— Radial—— Rectangular0 0.2 0.4 0.6 0.8 1 1.2depth, h/ri0.1—\ Circumferential--- RadialRectangularIIII00 0.2 0.4 0.6 0.8 1 1.242The above observations show that the radial and the circumferential strain gaugedesigns have opposite strengths and weaknesses. The radial design has higher sensitivityand the circumferential design has better non-uniform stress calculation stability. Thestrength of both designs can be combined by using a six-element rosette, as shown inFigure 21. This rosette has both radial and circumferential strain gauges. Eachcorresponding pair is connected in a half bridge circuit. This arrangement effectively addsthe strains of two gauges together because the circumferential strains have the oppositesign to the radial strains. Greatly increased thermal stability is an additional benefit.Figure 21 A layout of a 6-element combined radial and circumferential rosette.43Figure 22 shows strain sensitivity of the 6-element rosette, as indicated by thecalibration coefficients and b of the combined strain gauge is shown in . Calibrationcoefficient increases by more than 130 % and calibration coefficient b increases bymore than 60%. The increased strain sensitivity of the combined strain gauge isaccomplished without any cost on the part of the stress calculation accuracy of the methodfor measuring non-uniform residual stress measurement. To examine the improvement ofthis new 6-element rosette relative to other three designs, especially to the rectangularone, the graphs of the reliability factors of these rosettes are shown in Figure 23. Figure23 shows that the reliability factors for the 6-element rosette have greater values relativeto corresponding values for the rectangular rosette. This means that measuring residualstresses by a 6-element rosette gives more accurate results in the same depth range whicha rectangular rosettes gives reliable results. In addition, a 6-element rosette increases thepractical depth limit for measuring residual stress variation with depth. For example,considering the values of F for rectangular rosette at depth equal to 0.7r1 as the criterionfor acceptable accuracy, then as can be seen from Figure 23 the limit of measurementincreases to depth equal to 0.9r which is 30% improvement.440.81.20.80.60.4c.)0-0.2-0.4depth, h / sFigure 22 Calibration coefficient ä and b for four different strain gauge designs.0.60.40.2-0.2-0.4CombinedRadialflectanguir04 08 1.CircumferentialHdepth, h / i2Combined:Circumferential2450.1I.CICircumferentialRadial.-.. . ..::........ Rectangular. Combined0 0.2 0.4 0.6 0.8 1 1.2depth, h/riFigure 23 Reliability factors for four different strain gauge designs.depth, h I r0.080.060.040.020-0.020.140.120.10.08C0.060.020464.4 Experimental verificationTo verilr that the hole drilling method has a higher sensitivity using a 6-elementrosette than a conventional ASTM rectangular rosette, a set of experiments similar to theexperiments in Section 3.3 were conducted. One 6-element rosette and one rectangularRK-120 rosette were installed on a 12.7 x 38.1 x 565 mm aluminum bar. Similarprecautions as mentioned in Section 3.3 were conducted to contain undesirable bendingstrains within 2%. The bar was subjected to a range of tensile loads from zero to 35 kN in7 kN increments, and corresponding strains were recorded. The measured response of thestrain gauges per unit load was determined from the gradient of the strain versus loadplots. This procedure reduces the effect of random strain measurement errors and of anyexisting residual stresses.Holes were drilled at the centers of the two strain gauge rosettes in 14 consecutive0.20 mm depth increments. After drilling each increment, the sample was subjected to thesame range of loads and the corresponding strains were recorded. Again, the response ofthe strain gauges per unit applied load was determined from the gradient of the strainversus load plots. After the drilling, the diameters of the hole at the centers of 6-elementrosette and rectangular rosette were measured as 2.46 mm. From these data, thecalibration coefficients ã and b for a taper hole were evaluated using the methoddescribed by Rendler and Vigness [7].Figure 24 compares the experimentally determined calibration coefficients and bfor the 6-element rosette with the corresponding ones for the standard RK-120 rosette.These graphs confirm that the 6-element rosette improved over the standard RK-120rosette by more than 130% for calibration coefficient and more than 60% for thecalibration coefficients b. In addition, the theoretical values and b from finite elementcalculations for these two rosettes also are shown in Figure 24. The agreement between470.7CC1.21.080.20.60.50.40.30.20.100.00 0.50 1.00 1.50 2.00 2.50 3.00Hole depth, mm0.60.400.00 0.50 1.00 1.50 2.00 2.50 3.00Hole depth, mmFigure 24 Comparison of experimental and theoretical and b values for a 6-elementrosette and a rectangular rosette.48numerically calculated calibration coefficients and their values determined by experimentwere very good. The differences between experimental results and theoretical predictionsare less than 3% for the 6-element rosette and 4% for RK-120 rosette. Note that in thesegraphs, the hole depth is specified in millimeters rather than normalized with respect to themean rosette radius, r. This is done because the rm values for the 6-element rosette andthe standard rectangular rosette are different.49CHAPTER 5CONCLUSIONThe strain sensitivity and the accuracy of the hole-drilling method for measuringresidual stresses was investigated in this study. This investigation focused on the effectsof hole geometry and strain gauge rosette design. The study showed how these twofeatures could be modified to improve the strain sensitivity and accuracy of the method.This study demonstrated that the strain sensitivity can be improved significantly bymodifying the geometry of the hole. Using of a reverse taper hole instead of straight holewas found to increase strain sensitivity by between 20-80%. This increased strainsensitivity reduces the effect of strain measurement errors on the residual stresscalculation.The sensitivity increase is mostly due to increased flexibility of the reverse taper holecompared with a straight hole of the same surface diameter. This flexibility contributes tofaster stress relief in the area surrounding the hole. It also causes the strains around thehole to be relieved more rapidly which in turn make it possible to get the maximumpossible strain reliefs and consequently maximum strain sensitivity in a relatively shallowhole compared with the straight hole case. Therefore, a higher sensitivity is achievedwhile less damage is done to the specimen. The practicality of taper hole drilling wasdemonstrated by designing and building a special taper hole drilling jig. The jig combineddownward feed motion with and an off center orbiting.Experiments were conducted to verify the theoretical results obtained from the finiteelement calculations. Agreement within 3% was achieved when comparing the finiteelement calculated calibration coefficients and b for straight and taper holes with thecorresponding experimentally determined values. This confirms that the finite element50procedure is effective in determining the calibration coefficients for the hole-drillingmethod. The experiments also verified that the specially designed taper hole drilling jigwas able to drill reverse taper hole reliably. The jig was easy and convenient to use, andrequired only a small extra effort compared with straight hole drilling.The second part of the thesis investigates the effects of the strain gauge rosette designon the strain sensitivity and accuracy of the hole drilling method. The strain sensitivity ofradial and circumferential strain gauges was compared with that of conventionalrectangular strain gauges. Calibration coefficients for these three different designs wereevaluated. The relative merits of these designs for evaluating the variation of residualstresses with depth were investigated by studying their numerical conditioning andreliability factors.A radial strain gauge rosette showed a significant improvement in strain sensitivityrelative to the conventional rectangular design. Condition numbers of the radial straingauge, however, were worse than the corresponding condition numbers for therectangular strain gauge. The circumferential strain gauge, on the other hand, hasrelatively poor strain sensitivity, but has strong condition numbers. Because of its poorstrain sensitivity the circumferential strain gauge is not suitable for measuring residualstresses when used alone.A new 6-element rosette design is proposed which combines the beneficialcharacteristics of both radial and circumferential strain gauges. The new combined rosettehas the benefit of higher strain sensitivity and temperature compensation. This 6-elementdesign improves the calibration coefficients ä by more than 130% and the calibrationcoefficients b by more than 60%. The 6-element rosette also improves the accuracy ofthe incremental method for measuring non-uniform residual stress fields and increases themaximum allowable depth by 30%.51A set of experiments was conducted to verif,’ the higher strain sensitivity of the new6-element design in practice. These experiments confirmed the theoretical results that the6-element rosette has better strain sensitivity than the standard rectangular rosette, andthat the theoretical method realistically models practical experimental measurements.52REFERENCES(1) Mordfin, L. “Measurement of Residual Stresses: Problems and Opportunities,” inResidual Stress for Designers and Metallurgists, Vande Walle, L. J. (editor), 1981, pp.189-2 10(2) Measurements Group, Inc. “Measurement of Residual Stresses by the Hole-DrillingStrain Gauge Method” Tech Note TN-503-4, Measurements Group, Inc., Raleigh, NC,l9pp., (1993).(3) Mathar, J., “Determination of Initial Stresses by Measuring the Deformation aroundDrilled Holes,” trans., ASME 56, No. 4, 1934, pp. 249-254(4) Timoshenko, S. and Goodier, J. M. Theory ofElasticity, 3rd, ed., McGraw-Hill, NewYork, 1970.(5) Soete, W. and Vancrombrugge, R. “An Industrial Method for the Determination ofResidual Stresses,” Proceedings SESA, Vol. 8, (1), 1950, 17-28.(6) Kelsey, R. A. “Measuring Non-Uniform Residual Stresses by Hole Drilling Method,”Proceedings SESA, Vol. 14, (1), 1956, pp. 181-194.(7) Rendler, N. J. and Vigness, I. “Hole-drilling Strain-gage Method ofMeasuringResidual Stresses,” Experimental Mechanics, Vol. 6, No. 12, December 1966, pp. 577-586(8) American Society for Testing and Materials, “Standard Test Method for DeterminingResidual Stresses by the Hole-drilling Strain Gage Method,” ASTM Standard E837-92(1992).53(9) Beaney, E. M. and Procter, E. “A Critical Evaluation of the Center-Hole Techniquefor the Measurement of Residual Stresses,” Strain, Journal of BSSM Vol. 10, No. 1,1974, PP. 7-14.(10) Flaman, M. T., “Brief Investigation of Induced Drilling Stresses in the Center-HoleMethod ofResidual Stress Measurement,” Experimental Mechanics, Vol. 22, No. 1, 1982,pp. 26-30.(11) Schajer, G. S., “Application of Finite Element Calculation to Residual StressMeasurement.” Journal ofEngineering Materials and Technology, Vol. 103, April 1981,pp. 157-163(12) Schajer, G. S., “Measurement of Non-Uniform Residual Stresses Using the Hole-Drilling Method. Parts 1 and 2,” Journal ofEngineering Materials and Technology, Vol.110, October 1988, pp. 338-349(13) Schajer, G. S., “Use of Displacement Data to Calculate Strain Gauge Response inNon-uniform Stress Fields,” Strain, Vol. 29, No. 1, 1993. pp. 9-13(14) Bijak-Zochowski, M. “A Semi-destructive Method of Measuring Residual Stresses,”VDI-Berichte, Vol. 313, pp.469-76.(15) Flaman, M. T. and Manning, B. H., “Determination of Residual Stress Variation withDepth by Hole-Driffing Method,” Experimental Mechanics, Vol. 25, No. 9, 1985, pp.205-207.(16) Flaman, M. T., IVlills, B. E., and Boag, J. M., “Analysis of Stress Variation withDepth Measurement Procedures for Center-Hole Method Residual Stress Measurement,”Experimental Techniques, Vol. 11, No. 6, 1987, pp. 3 5-37.54APPENDIXCalibration coefficients for a circumferential strain gaugeAt each point of a single arc filament with the width w, radius r, and arc angle ii6the circumferential strain is:u dv— +r rd6This strain changes the resistance dR in a infinitesimal arc d8 as:z dR=F 8 dR=Fr-60Integrating over the arc lengthAR= FprjB2 I+-i8 (Al)w 0 r rd9}For calibration coefficient A the displacement field is symmetrical, therefore for the ithfilament:AR. = FArIu$02d9 Fpu Aw r°i wTotal change of resistance for n filaments is:ART =AR =A 8u,i=1 1=155But ART = F RT8, where RT is the total original resistance of the grid and 6 is averagestrain measured over the strain gauge. But RT= Fp tX9r. Substituting in the abovewEquation gives:u. u.1= A= (A2)2rIn the case of evaluating B: u. = U(rjcos26 and v1 = V(i)sin2& then from Equation(Al):AR. FPrfB2 I--+-5-Id8= Fpç f02 I-cos26dO+!dv.1w Lr r1dO) wr u. ÷2vsinAOcos2q = (u1+2v)sinA8cos2çow 2r1 wwhere o is angle between the mid radial axis and the principal stress direction. Then Bcan be calculated as follow:ART = = sinAOcos2ç(U +2)= T6i=1 w i=1=— (U+2VJsinA9cos2ço8-r1A6= B=(U+2VjsjnAO (A3)2cos2ç A956

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