RESISTANCE STRAIN GAGES FILLAMENTS EFFECT

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1 RESISTANCE STRAIN GAGES FILLAMENTS EFFECT Nashwan T. Younis, Department of Mechanical Enineerin, Indiana University-Purdue University Fort Wayne, USA Bonsu Kan, Department of Mechanical Enineerin, Indiana University- Purdue University Fort Wayne, USA ABSTRACT The subject of error usin strain aes is complex and it is not the intension to present a comprehensive analysis. In this study, the discrete averain effects of a strain ae alon the ae filaments are taken into account in the assessment of the errors due to placement of the ae, ae lenth, ae width, and number of filaments. The ae is placed near the ede of a hole in an infinite plate subjected uniaxial tension. It is shown that the averae strain over the ae filaments is not the same as the averae strain over the ae rid area. Recommendations for selectin aes are presented. Keywords: Experimental stress analysis, Strain ae INTRODUCTION Most people come across strain ae based devices on a daily basis but unaware of the fact. Experimental stress analysis enineers have both personal and professional interaction with strain aes and perhaps are no more aware of the backround about the manitude of averain effects of aes or the principle of strain aes. In 6, Robinson discussed the fact that stress analysis enineers probably no more aware of the backround reardin some aspects of the aes than the averae person [1]. Solvin touh strain ae problems becomin what was old and unsolved is new aain. Cappa et al. [] developed a conditionin unit based on the direct resistance measurement method by drivin a constant current throuh a strain ae. Ajovalasit s [3] analysis showed the output of a strain ae is influenced by the coupled effect of transverse sensitivity and pressure sensitivity on the ae. An experimental method was developed to determine the stiffness of some commercial strain aes [4]. The results showed that strain ae stiffness is a feature to be considered for the evaluation of the local and lobal reinforcement effects. Strain aes are accurate when the ae is placed on isotropic materials with relatively uniform stress distributions. However, the accuracy of the strain ae is not always hih. Error is introduced from the physical nature of the strain ae and how it measures the strain. The error is acceptable, but increases sinificantly when the ae is used to measure strain at discontinuities such as holes, notches, and crack tip; areas of steep strain radients. In the field of fracture mechanics, some researchers use the ae readin as the strain at the center of the ae. This is not an accurate representation of the stress field at a crack tip due to the averain effect. Irwin suested that mode

2 one stress-intensity factor, K I, near a crack tip can be determined experimentally by usin strain aes [5]. In 1987 Dally and Sanford developed a method that utilizes strain aes for determinin K I [6]. An overdeterministic approach was developed for measurin K I usin data from three 1-element strip strain aes with some accuracy [7]. The error enerated in K I due to placement errors in positionin and orientin the sinle ae was determined [8]. It was concluded that the deviation in the orientation anle of the strain ae is the dominant source of the error. A technique for stress intensity factor determination usin strain aes was developed in which the location of the strain ae relative to the crack tip was chosen throuh parametric study of the asymptotic fields [9]. Sarani et al. [1] proposed a finite element based method for determinin the limitin radial distance of placin the strain ae in the vicinity of a crack tip. Recently, the effect of strain radient was mentioned or inored by researchers in the field. To assure accuracy of determinin structural ae sensitivity, Zhan et al. chose locations for the strain ae rosettes with small strain radient [11]. Hole-drillin method is the most common technique for measurin residual stresses in various materials and structures [1]. It utilizes strain aes around circular hole. The averain effect of the ae around the hole is inored, perhaps due to the difficulty in estimatin the manitude of the error. The correction of errors introduced by hole eccentricity has been proposed by Barsanescu and Carlescu [13]. In 9, the performance of three dimensional strain aes embedded into a sphere was evaluated [14]. In the case of hih strain radient, the calculated strain tensor is subject to errors and this problem can be overcome by embeddin the aes in a sphere at a specified orientation relative to the center of the sphere. The subject of error in a strain measurement system usin metal-foil sensin rid is complex. In 1984, Pople listed the human factor errors and error sources in strain ae measurement [15]. Perry presented an extensive report that examined several fundamental properties of the strain ae that are involved in measurement accuracy [16]. These properties are the ae factor, reinforcement effects, transverse sensitivity, and thermal effects. In addition, Perry plotted the percentae difference between the peak strain at a hole and the strain interated over the ae square area aainst the ratio of the rid area to hole radius. The traditional method of examinin the strain averain is to use the averae strain over the area of the entire strain ae rid. The ae filament covers only a portion of this area, and this fact needs to be accounted for in an analysis of errors due to strain ae placement. The averae strain over the ae filaments is not the same as the averae strain over the ae rid area. In a series of articles deal with resistance strain aes, Stein shared his experience that he developed over more than forty years [17]. The articles are about transducers and sinal conditionin as strain aes are made of resistive filaments. Strain aes are available in many different sizes and confiurations. Therefore, understandin the

3 variables in the averain effect will result in proper ae selection. In this paper, a comprehensive analysis of the discrete averain effect associated with hih strain radient is presented. Recommendations are made for selectin aes for hih strain radient measurements at the end. The strain averain is modeled with the intention of extractin sufficient knowlede to illustrate the typical results that can be expected from strain aes at stress concentration areas. A classical plate with a circular hole subjected to uniaxial loadin is used as a vehicle for examinin the effects of ae lenth, ae width, number of filaments, and ae misplacement on the averae strain experienced by the ae. THEORY The strain ae is a type of electrical resistor. Most commonly, strain aes are thin metal-foil rids that are bonded to the surface of a machine part or a structural member. When forces are applied to the member, the ae elonates or contracts with the member, creatin normal strains. The chane in lenth of the ae alters its electrical resistance. By measurin the electrical resistance of the wire, the ae can be calibrated to read values of normal strain directly. Since the ae is of finite lenth, the chane in resistance is due to the averae strain alon the ae and not the center strain in eneral. If the strain alon the ae is constant or linear, the averae strain is the same as that of the center strain. However, for a stress concentration problem, the averae strain will differ from the center strain. When the strain radient is lare, the averae strain is lower than the true strain at a point. Therefore, the indicated strain will be in error of the true strain. The photoetchin process used to create the metal-foil rids is very versatile, enablin a wide verity of ae sizes and rid shapes to be produced [18]. The rid is bonded to a thin plastic backin film or carrier because the foil is fraile. The backin provides three main functions: 1. Means of handlin the foil durin the installation process of the strain ae. Bondable surface for adherin the ae to the specimen 3. Electrical insulation between the ae and the object bein tested 4. Space for alinment markins and solder tabs to attach the lead wires It is clear that the thin plastic backin is an interal part to the basic ae construction. However, conformance to measure peak strain in the vicinity of stress concentration will be reduced. Each ae consists of a fine metal rid that is stretched or shortened when the object is strained at the area where the ae is attached. The rid is equivalent to a continuous wire that oes back and forth from one end of the rid to the other, therefore effectively increasin its lenth. The rid of bonded foil ae, shown in Fi. 1, is the major source of error introduced by the strain ae when it is used to measure strain at the ede of the hole. The electrical resistance strain ae measures the averae strain of each filament and the indicated strain is the averae of the filaments strains. Usin the points shown in Fi. 1, the indicated strain is:

4 Y rr r r B C P F G a Y c L X X c A D W E H Fiure 1. Definition of the Strain Gae Geometry. indicated AB CD EF GH (1) number of filaments One of the most important problems in the desin of plate structures is to determine the stress concentration due to the presence of holes and other discontinuities. The classical Kirsch [19] solution for the stresses around a circular hole in a lare plate with normal stress applied at infinity in the Y- direction, as shown in Fi. 1, is iven by: a a rr cos r r (.1) 4 a a cos (.) 4 r r a a sin (.3) r r r In the end, the strain averain of the ae is computed by interatin the strains alon the ae filaments, for which the strain in the Y-direction is needed alon each filament. This requires convertin the stresses in the r- coordinate system to ones in the X-Y coordinate system. Upon applyin the standard stress transformation equations, the stresses at point P can be found by: cos sin cos sin (3.1) X Y rr rr r sin cos cos sin (3.) r or, since r X Y and tan 1 ( Y X ), r r r r r r r r 4 x x x x y r r r r r r r r 4 y x y x x y (4.1) (4.)

5 .5 (a).4.3 y y.1 y. y.3 y.4 x y x 3. (b).5 y y.1 y y.. y. 3 y.4 y x Fiure. Distribution of (a) σ x /σ o and (b) σ y /σ o Alon the x-axis Around the Hole s Boundary for =.3. Note that the X- and Y-coordinate have been normalized aainst the hole s radius; i.e., x=x/a, y=y/a, and r x y. The distribution of σ x /σ o is plotted as a function of position alon the x axis for various values of y in Fi. a. An examination of this fiure clearly indicates that the transverse stress is zero at the ede of the hole (x=) and varies in different patterns as y increases from to.5. Thus, there is no area to mount the ae on that will represent the averae σ x. A perfect ae installation would be a x=1 when

6 the ae filaments are perpendicular to the x axis. To account for the expertise of the user (tiltin the ae), the distributions at x less than 1 is plotted. With the hole center at the oriin, the variation of σ y /σ o of Eq. (4.) in the x direction at y=,.1,.,.3,.4,.5 is plotted in Fi. b. Examinin the stress curves, one can see that the slopes chane not only in manitude, but also in sin, from neative to positive and vice versa. Stress is a mathematical abstraction, and it can not be measured. Strains, on the other hand, can be measured directly throuh well-established experimental procedures such as strain aes. Once the strains in a component have been measured, the correspondin stresses can be calculated usin stressstrain relationships such as eneralized Hook s law. However, in this paper, the strains need to be calculated utilizin the stress equations. The stress-strain relations for a two-dimensional state of stress are: x 1 E x y y 1 (5.1-) y x E Shown in Fi. 3 is the distribution of y in the first quadrant of the x-y plane for σ y /E=1 and =.3. It can be seen that the maximum value is 3 at x=1 and y=, which is well known as the maximum stress concentration factor σ y /σ o for the present case. Note also that, althouh not clearly shown in the raph, y = at x= and y=1 since σ x /σ o =1 and σ y = at that point. The results show that the averae strain does not equal the strain at the ae center. The traditional method of examinin the strain averain is to use the averae strain over the area of the entire strain ae rid. However, the ae filament covers only a portion of this area as shown in Fi. 1. This fact needs to be accounted for in an analysis of errors due to strain ae placement. The strain averain of the ae is modeled by interatin the strains alon the filaments. Thus, the strain in the y direction is needed alon each filament. The last step is to compute the averae strain experienced by n filaments placed in the y-direction, which can be found by: n 1 av yk dy (6) nl l 1 k where l =L /a, the normalized ae s lenth, and yk denotes the strain in the k th filament. The discrete averain effects of a ae in the vicinity of a hole are function of the dimensions of the ae. It is advantaeous to compare the size of the ae to the hole radius a. In the examples presented, it is assumed that the ede of the ae (or the first filament) is at distance x c =X c /a from the ede of the hole and the vertical center of the ae is displaced from the lateral axis of the hole by y c =Y c /a, as shown in Fi 1. The width of the ae is w=w/a and the filaments are evenly spaced across the width. The averae strain now becomes a function of these variables and written as: f ( w, l, x, y, n) (7) av c c

7 y x Fiure 3. y in the First Quadrant of the x-y Plane. /E=1 and =.3. RESULTS AND DISCUSSION It is assumed that no shear la occurs across the adhesive line, thus the strain felt by each ae filament is the same as the strain in the plate directly below it. The strain is averaed over each filament lenth. It is also assumed that the strain does not vary across an individual filament and the strains at the midline of the filaments are used in the averain process. The filaments are evenly spaced across the width. Equations (6) and (7) provide a settin for evaluatin the influences of each of the 5 parameters considered in this study. A few examples are presented to illustrate the manitudes of error that are possible. The percent error e is defined with respect to the maximum normal strain that occurs at x=1 and y=. In addition, the averae strain av within the area A covered by the ae, which corresponds to the mathematical averae strain for n=, is introduced for comparison: 1 n nl av lim n k 1 l yk dy 1 da, A A y A l w (8) Case 1: In the first case the ae lenth is held constant while the ae width is varied with. Both Xc and Yc are zero. Fi. 4 shows the variation of the averain strain with the W/a for different number of filaments n. Additional filaments further away from the hole increases the error for a iven width

8 ae. A sinle element at hole s ede ives an error of 7.4%. It is important to remember that these curves are for the same model, i.e. the same hole, material, and applied load, yet the strain is function of the ae width and the number of filaments. For W/a=.5, the percent difference of the error between n= and n=1 is 16.6% while the percent difference of the error between n= and n=1 for W/a of 1. is 5.3%. One can conclude that the spacin between the filaments of a strain ae contributes to the error in measurin the strain. However, as an example for n=, the percent difference of the error between W/a=.5 and W/a=1. is 14.5%. Thus, the number of filaments is the dominant factor in the assessment of the error in strain ae measurements at the vicinity of stress concentration reion. As a rule of thumb, the ae size should be very small as compared to the hole size. However, small strain aes tend to exhibit deraded performance in terms of the maximum allowable elonation, the stability under static strain, and the endurance when subjected to alternatin cyclic strain []. Case : In this case, the ae width is kept constant and the ae lenth varied. The percentae error versus L /a for different number of filaments is shown in Fi. 5. It is clear that the averae strain is a function of the ae lenth. However, for a iven load the strain at the ede of the hole is constant. The difference in error between a sinle filament and two filaments is lare for the same L /a ratios. It is clear that the multiple filaments tend to the same line for lare L /a ratios. The results indicate that for L /a=.5 the percent difference of the averae strain between n= and n=1 is 16.7%. The percent difference in the averae strain between L /a=.5 and L /a= for n= is 35%. Therefore, both the ae lenth and the number of filaments should be considered in the assessment of the error of the averae strain in the vicinity of stress concentration reion. The user can select the smallest practicable ae n= n=3 n=5 n=1 n= e [%] 3 1 n= w Fiure 4. Effect of Gae s Size on Error Versus Width (w=w/a) for l =.5 and x c =y c =.

9 6 5 n= n=1 4 e [%] 3 1 n= n=3 n=5 n= l Fiure 5. Effect of Gae s Size on Error Versus Lenth (l =l /a) for w=.5 and x c =y c =. lenth, but has to be aware of the reatly increased error and uncertainty in the indicated strains due to the nature of the ae and instrumentations. Case 3: In this case, both the ae lenth and width are held constant while the horizontal distance from the hole s ede to the first filament, X c, varied. This is due to the facts that the matrix width of the strain ae is reater than the rid width and human-dependant error source. The metal-foil strain ae is the most frequently employed ae for both eneral-purpose stress analysis and transducer applications. The rids are very fraile and easy to distort, wrinkle, or tear. For this reason, foil aes are enerally mounted on a thin epoxy carrier or paper or sandwiched (encapsulated) between two thin sheets of epoxy; this improves the temperature rane, fatiue life, and chemical mechanical protection of the sensin rid. The dimensions of the matrix (sheets) are larer than that of the sensin rid. Hence, obtainin X c = in enineerin practice is very difficult. The results are plotted in Fi. 6. The percent difference of the error, for X c =, between n= and n=1 is 18.3%. But, for n=, the difference of error between X c /a= and X c /a=.5 is 47.5%. This clearly shows that the predominant factor in the error of the averae strain is the lateral misdisplacement of the ae near the ede of the hole. The results also indicate that the number of filaments contributes sinificantly to the error of the strain measured at a stress concentration area. Case 4: This case deals with the misplacement of the strain ae which is a typical human-dependant error. The distance between the hole s x axis and the ae horizontal center line is Y c. Obtainin Y c = in most practical situations is very difficult even for experienced operator with considerable skill and aility. Fi. 7 shows the strain averain effect versus Y c /a for W=L =.5. The results show that the

10 6 5 n= n=1 4 e [%] 3 1 n= n=3 n=5 n= x c Fiure 6. Effect of Gae s Alinment on Error Versus Lateral Alinment (x c =X c /a) for w=.5 and l = n= n=1 e [%] 3 1 n= n=3 n=5 n= y c Fiure 7. Effect of Gae s Alinment on Error Versus Vertical Alinment (y c =Y c /a) for w=.5 and l =.5. number of filaments and the ae mispositionin contribute to the error of the averae strain. For example, the percent difference of the averae strain between n= and n=1 for perfect ae alinment is 16.5%. The difference of the averae strain between properly alined ae (n=) and a ae displaced from the lateral axis of the hole by y c =.3 is 4% for n=. From the results obtained, the error of measurin the strain at a the ede of a hole usin strain aes can be clearly shown Fis Each line on the raphs represents the strain experienced by the ae for a iven number of filaments.

11 As the number of filaments chane, the strain also chanes. Hence, the strain readin is a function of the number of filaments in the rid. The strain in the vicinity of a hole is constant at fixed applied load, but the strain ae results depend on the number of filaments. This shows that there is an error in the strain ae measurement and the number of filaments has a lare influence on the averae strain near a stress concentration area. CONCLUSIONS In this study, results of discrete averain effects of a strain ae near a hole were presented. Four separate cases were investiated. Case 1 involved the effect of the ae width and case examined the effect of the ae lenth. Cases 3 and 4 dealt with the mispositionin of the strain ae. In each case, the effects of the number of filaments upon the averae strain were studied. The existence of error can be seen by examinin the variables of cases 1,, 3 and 4. When measurin the strain at the ede of a hole, the ae lenth and width present definite error. If these variables did not affect the strain averae, the lines in Fi 4 would be horizontally linear. This is definitely not the case. Therefore, the manitude of the strain ae readin error is dependent on the ae dimensions and position. The eneral trend of the strain ae readin (the averae strain) is less than the actual strain as the distance from the hole increases. In addition, the averae strain will decrease as the width of ae, lenth of the ae, or number of filaments in the rid increase. Therefore, the strain ae readin will underestimate the true strain at the ede of the hole. Thus, the number of ae filaments should be reduced in the hole-drillin method. The results can be used as a uide in correctin the measured strain at any circular hole. Finally, the averae strain over the ae filaments is not the same as the averae strain over the ae rid area. This fact needs to be accounted for in determinin stress concentrations usin strain aes. The results show the error can be reduced sinificantly if an experimental stress analysis enineer selects a ae with a few filaments. There are at least of filaments in most commercial strain aes. Therefore, a special purpose strain aes with few filaments should be manufactured and used for many practical applications. REFERENCES 1. Robinson, M., (6), Strain Gae Materials Processin, Metallury, and Manufacture, Experimental Techniques 3, Cappa, P., Del Prete, Z., and Marinozzi, F., (1), Lon Term Stability of a Novel Strain Gae Conditioner Based on the Direct Resistance Method, Experimental Techniques 5, Ajovalasit, A., (5), Embedded Strain Gaues: Effect of the Stress Normal to the Grid, Strain 41, Ajovalasit, A., D Acquisto, L., Fraapane, S., and Zuccarello, B., (7), Stiffness and

12 Reinforcement Effect on Electrical Resistance Strain Gaues, Strain 43, Irwin, G. R., (1957), Analysis of Stresses and Strains Near the end of a Crack Traversin a Plate, Applied Mechanics Dally, J. W. and Sanford, R. J., (1987), Strain-Gae Methods for Measurin the Openin-Mode Stress-Intensity Factor, K I, Experimental Mechanics 7, Berer, J. R. and Dally, J. W., (1988), An Overdeterministic Approach for Measurin K I Usin Strain Gaes, Experimental Mechanics 8, Berer, J. R. and Dally, J. W., (1988), An Error Analysis for a Sinle Strain-Gae Determination of the Stress-Intensity Factor K I, Experimental Techniques 1, Marur, P. R. and Tippur, H. V., (1999), A Strain Gae Method for Determination of Fracture Parameters in Bimaterial Systems, Enineerin Fracture Mechanics 64, Sarani, H., Murthy, K. S. and Chakraborty, D., (1), Radial Locations of Strain Gaes for Accurate Measurement of mode I Stress Intensity Factor, Materials & Desin, in press. 11. Zhan, S. Y., Prater, Jr., G., Shahhosseini, A. M., and Osborne, G. M., (8), Experimental Validation of Structural Gae Sensitivity Indices for Vehicle Body Structure Optimization, Experimental Techniques 3, Chen, J., Pen, Y., and Zhao, S., (9), Comparison Between Gratin Rosette and Strain Gae Rosette in Hole-Drillin Combined Systems, Optics and Lasers in Enineerin 47, Barsanescu, P. and Carlescu, P., (9), Correction of Errors Introduced by Hole Eccentricity in Residual Stress Measurement by the Hole-Drillin Strain-Gae Method, Measurement 4, Trench, P., Little, E. G., Tocher, D., O Donnell, P., and Lawlor, V., (9), The Performance of Three-Dimensional Strain Rosettes Evaluated when Embedded into a Sphere, Strain 45, Pople, J., (1984), Error in Strain Measurement-the Human factor (or how much do I contribute?), Experimental Techniques 8, C. C. Perry, C. C., (1984), The Resistance Strain Gae Revisited, Experimental Mechanics 4, Stein, P. K., (1999), Back to Basics, Experimental Techniques 3, Philpot, T. A., (8), Mechanics of Materials: An Interated Learnin System, John Wiley, USA, (8). 19. Dally, J. W. and. Riley, W. F., (1991), Experimental Stress Analysis, McGraw-Hill, New York, USA.. Strain Gae Selection, Criteria, Procedures, Recommendations. Tech. Note TN-55-1, Measurements Group, Inc., Raleih, NC.