# ASTM E1561-93 (Reapproved 2014)

Designation: E1561 − 93 (Reapproved 2014)Standard Practice forAnalysis of Strain Gage Rosette Data1This standard is issued under the fixed designation E1561; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (´) indicates an editorial change since the last revision or reapproval.INTRODUCTIONThere can be considerable confusion in interpreting and reporting the results of calculationsinvolving strain gage rosettes, particularly when data are exchanged between different laboratories.Thus, it is necessary that users adopt a common convention for identifying the positions of the gagesand for analyzing the data.1. Scope1.1 The two primary uses of three-element strain gagerosettes are (a) to determine the directions and magnitudes ofthe principal surface strains and (b) to determine residualstresses. Residual stresses are treated in a separate ASTMstandard, Test Method E837. This practice defines a referenceaxis for each of the two principal types of rosette configura-tions used and presents equations for data analysis. This isimportant for consistency in reporting results and for avoidingambiguity in data analysis—especially when computers areused. There are several possible sets of equations, but the setpresented here is perhaps the most common.2. Referenced Documents2.1 ASTM Standards:2E6 Terminology Relating to Methods of Mechanical TestingE837 Test Method for Determining Residual Stresses by theHole-Drilling Strain-Gage Method3. Terminology3.1 The terms in Terminology E6 apply.3.2 Definitions of Terms Specific to This Standard:3.2.1 reference line—the axis of the a gage.3.3 Symbols:3.3.1 a, b, c—the three-strain gages making up the rosette.3.3.1.1 Discussion—For the 0° – 45° – 90° rosette (Fig. 1)the axis of the b gage is located 45° counterclockwise from thea (reference line) axis and the c gage is located 90° counter-clockwise from the a axis. For the 0° – 60° – 120° rosette (Fig.2) the axis of the b gage is located 60° counterclockwise fromthe a axis and the c axis is located 120° counterclockwise fromthe a axis.3.3.2 εa, εb, εc—the strains measured by gages a, b, and c,respectively, positive in tension and negative in compression.3.3.2.1 Discussion—After corrections for thermal effectsand transverse sensitivity have been made, the measuredstrains represent the surface strains at the site of the rosette. Itis assumed here that the elastic modulus and thickness of thetest specimen are such that mechanical reinforcement by therosette are negligible. For test objects subjected to unknowncombinations of bending and direct (membrane) stresses, theseparate bending and membrane stresses can be obtained asshown in 4.4.3.3.3 εa ,εb , εc —reduced membrane strain components (4.4).3.3.4 εa“, εb“, εc“—reduced bending strain components (4.4).3.3.5 ε1—the calculated maximum (more tensile or lesscompressive) principal strain.3.3.6 ε2—the calculated minimum (less tensile or morecompressive) principal strain.3.3.7 γM—the calculated maximum shear strain.3.3.8 θ1—the angle from the reference line to the directionof ε1.3.3.8.1 Discussion—This angle is less than or equal to 180°in magnitude.3.3.9 C, R—values used in the calculations. C is thelocation, along the ε-axis, of the center of the Mohr’s circle forstrain and R is the radius of that circle.1This practice is under the jurisdiction of ASTM Committee E28 on MechanicalTesting and is the direct responsibility of Subcommittee E28.01 on Calibration ofMechanical Testing Machines and Apparatus.Current edition approved April 15, 2014. Published August 2014. Originallyapproved in 1993. Last previous edition approved in 2009 as E1561–93(2009). DOI:10.1520/E1561-93R14.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at service@astm.org. For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States14. Procedure4.1 Fig. 3 shows a typical Mohr’s circle of strain for a0° – 45° – 90° rosette. The calculations when εa, εb, εc, aregiven are:C 5εa1εc2(1)R 5 =~εa2 C!21~εb2 C!2(2)ε15 C1R (3)ε25 C 2 RγM5 2Rtan 2θ15 2 ~εb2 C!/εa2 εc(4)4.1.1 If εbC, then the ε1-axis is counterclockwise fromthe reference line.4.2 Fig. 7 shows a typical Mohr’s circle of strain for a0° – 60° – 120° rosette. The calculations when εa, εb, εc, aregiven are:C 5εa1εb1εc3(5)R 5 =2/3@~εa2 C!21~εb2 C!21~εc2 C!2# (6)ε15 C1R (7)ε25 C 2 RγM5 2Rtan 2θ15~εb2 εc!=3~εa2 C!(8)4.2.1 If εc− εb0, then the ε1-axis is clockwise from thereference line (see Note 1).4.3 Identification of the Maximum Principal Strain Direc-tion:4.3.1 Care must be taken when determining the angle θ1using (Eq 10)or(Eq 14) so that the calculated angle refers tothe direction of the maximum principal strain ε1rather than theminimum principal strain ε2. Fig. 10 shows how the doubleangle 2θ1can be placed in its correct orientation relative to thereference line shown in Fig. 1 and Fig. 2. The terms “numera-tor” and “denominator” refer to the numerator and denominatorof the right-hand sides of (Eq 10) and (Eq 14). When bothnumerator and denominator are positive, as shown in Fig. 10,the double angle 2θ1lies within the range 0° ≤ 2θ1≤ 90°counterclockwise of the reference line. Therefore, in thisparticular case, the corresponding angle θ1lies within the range0° ≤θ1≤ 45° counterclockwise of the reference line.FIG. 1 0° – 45° – 90° RosetteFIG. 2 0° – 60° – 120° RosetteFIG. 3 Typical Mohr’s Circle of Strain for a 0° – 45° – 90° Ro-sette FIG. 4 Differential Element on the Undeformed SurfaceE1561 − 93 (2014)24.3.2 Several computer languages have arctangent functionsthat directly place the angle 2θ1in its correct orientation inaccordance with the scheme illustrated in Fig. 10. Whenworking in Fortran or C, the two-argument arctangent func-tions ATAN2 or atan2 can be used for evaluating (Eq 10) and(Eq 14).4.4 Interpretation of Maximum Shear Strain—Ordinarily thesense of the maximum shear strain is not significant whenanalyzing the behavior of isotropic materials. It can, however,be important for anisotropic materials, such as composites.Mohr’s circle for strain can be used for interpretation of thesense of the shear strain. Fig. 3 shows a typical circle for a0°–45°–90° rosette. A differential element along and perpen-dicular to the reference line is initially as shown in Fig. 4. Itsdeformed shape, corresponding to the assumed strains, isshown in Fig. 5. The planes of maximum shear strain are at 45°to the θ1direction as in Fig. 6 (see Note 2).FIG. 5 Deformed Shape of Differential ElementFIG. 6 Planes of Maximum Shear StrainFIG. 7 Typical Mohr’s Circle of Strain for a 0° – 60° – 120° Ro-setteFIG. 8 Gage Labeling for Back-to-Back RosettesFIG. 9 Gage Labeling for Back-to-Back RosettesFIG. 10 Correct Placement of the Double Angle 2 θ1E1561 − 93 (2014)34.5 Back-to-Back Rosettes:4.5.1 When the loading of a member or structure mayintroduce bending strains in the surface at the intended site ofthe rosette, back-to-back rosette installations are commonlyemployed, as shown in Fig. 8 and Fig. 9, to permit separatedetermination of the bending and membrane strains.4.5.2 When rosettes are used on both sides of thin materials,the labeling alternatives are:4.5.2.1 Label as in Fig. 8, which follows the sign conventionof Fig. 1 and Fig. 2 as the observer faces each of the rosettes.4.5.2.2 Label, for example, the gage on face 1 in thecounterclockwise direction and the gage on face 2 in theclockwise direction, both as seen by an observer facing therosette (see Fig. 9).4.5.2.3 Labeling (4.5.2.1) requires no sign change in thedata reduction equations or in the interpretation of the angles.Results are still interpreted as the observer faces the rosette.4.5.2.4 Labeling as described in 4.5.2.2, wherein the ob-server fixes the a legs of the rosettes on both sides of the plateor skin to coincide in direction, is particularly convenient forthe separation of bending and membrane strains. It also reducesthe likelihood of a wiring or computational error which mayoccur in converting from the labeling in 4.5.2.1 to accomplishthe basic purpose of back-to-back rosette installations. Thefollowing procedure is limited to test materials which arehomogeneous in the thickness direction, or are symmetricallyinhomogeneous with respect to the midpoint of the thickness,as in many laminated composite materials.NOTE 1—The equations in 4.1 and 4.2 are derived from infinitesimal(linear) strain theory. They are very accurate for the low strain levelsnormally encountered in the stress analysis of typical metal test objects.They start to become detectably inaccurate for strain levels greater thanabout 1 %. Rosette data reduction for large strains is beyond the scope ofthis guide.NOTE 2—The Mohr’s circle for strain is constructed in generally thesame manner as the Mohr’s circle for stress. Normal strains, ε, are plottedas abscissae-positive for elongation and negative for contraction. One-halfthe shear strains, γ/2, are plotted as ordinates. If the shear strains onopposite sides of an element of area appear to form a clockwise couple,then γ/2 is plotted on the upper half of the axis. Similarly shear strainswhich appear to form a counterclockwise couple plot on the lower half.With this convention, angular directions on the circle are the same asangular directions on the specimen. See Fig. 3.4.6 In those cases where the gages are not wired toautomatically cancel the bending components of strain withinthe Wheatstone bridge circuit, the following relationships canbe employed with the rosette labeling in Fig. 9 to separatelydetermine the membrane and bending strain components.4.6.1 For the membrane components of the strain (that is,the through-the-thickness uniform strains, after removing thesuperimposed bending strains):εa 5 ~εA1εA1!/2 (9)εb 5 ~εB1εB1!/2 (10)εc 5 ~εC1εC1!/2 (11)4.6.2 For the bending components of strain, at both surfacesof the test object:εa“56~εA2 εA1!/2 (12)εb“56~εB2 εB1!/2 (13)εc“56~εC2 εC1!/2 (14)where:εa, εb, εc= reduced membrane strain components in the di-rections of the three rosette legs when labeled inaccordance with Fig. 9.εa“, εb“, εc“= reduced bending strain components in the direc-tions of the three rosette legs when labeled inaccordance with Fig. 9.4.6.3 The strain terms in (Eq 15) through (Eq 20) withcapitalized subscripts represent the measured strains (aftercustomary corrections) from the corresponding rosette legs asshown in Fig. 9.5. Report5.1 The rosette data analysis may be part of the report on atest program. Report the following information:5.1.1 Description of gages and measuring equipment,5.1.2 Location and orientation of strain gage rosette,5.1.3 Measured strains (corrected), and5.1.4 Calculation of principal strains.6. Keywords6.1 bending strain; Mohr’s circle for strain; rosette; shearstrain; strain; strain gages; tensile strainASTM International takes no position respecting the validity of any patent rights asserted in connection with any item mentionedin this standard. Users of this standard are expressly advised that determination of the validity of any such patent rights, and the riskof infringement of such rights, are entirely their own responsibility.This standard is subject to revision at any time by the responsible technical committee and must be reviewed every five years andif not revised, either reapproved or withdrawn. Your comments are invited either for revision of this standard or for additional standardsand should be addressed to ASTM International Headquarters. Your comments will receive careful consideration at a meeting of theresponsible technical committee, which you may attend. If you feel that your comments have not received a fair hearing you shouldmake your views known to the ASTM Committee on Standards, at the address shown below.This standard is copyrighted by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959,United States. Individual reprints (single or multiple copies) of this standard may be obtained by contacting ASTM at the aboveaddress or at 610-832-9585 (phone), 610-832-9555 (fax), or service@astm.org (e-mail); or through the ASTM website(www.astm.org). Permission rights to photocopy the standard may also be secured from the Copyright Clearance Center, 222Rosewood Drive, Danvers, MA 01923, Tel: (978) 646-2600; http://www.copyright.com/E1561 − 93 (2014)4