AE3145 Strain Transformation and Rosette Gage Theory Page 1Strain Transformation and Rosette Gage TheoryIt is often desired to measure the full state of strain on the surface of a part, that is to measurenot only the two extensional strains, εx and εy, but also the shear strain, γxy, with respect to somegiven xy axis system. It should be clear from the previous discussion of the electrical resistancestrain gage that a single gage is capable only of measuring the extensional strain in the directionthat the gage is oriented. Assuming that the x and y axes are specified, it would be possible tomount two gages in the x and y directions, respectively to measure the associated extensionalstrains in these directions. However, there is no direct way to measure the shear strain, γxy. Noris it possible to directly measure the principal strains since the principal directions are notgenerally known.The solution to this problem lies in recognizing that the 2D state of strain at a point (on asurface) is defined by three independent quantities which can be taken as either: (a) εx, εy, andγxy, or (b) ε1, ε2, and θ, where case (a) refers to strain components with respect to an arbitrary xyaxis system, and case (b) refers to the principal strains and their directions. Either case fullydefines the state of 2D strain on the surface and can be used to compute strains with respect toany other coordinate system. This situation implies that it should be possible to determine these3 independent quantities if it is possible to make three independent measurements of strain at apoint on the surface. The most obvious approach is to place three strain gages together in a“rosette” with each gage oriented in a different direction and with all of them located as closetogether as possible to approximate a measurement at a point. As will be shown below, if thethree strains and the gage directions are known, it is possible to solve for the principal strains andtheir directions or equivalently, the state of strain with respect to an arbitrary xy coordinatesystem. The relations needed are the strain transformation equations and Mohr’s Circleconstruction provides a good visualization of this process.2D Strain Transformation and Mohr’s CircleThe two dimensional strain transformation equations were developed in earlier structuralmechanics courses and are very similar to the 2D stress transformation equations (see forexample, chapter 6 in, Gere & Timoshenko, Mechanics of Materials, 3rd edition, 1990). Withoutrederivation, they are repeated below:)sin(coscossin)(2cossincossincossinsincos222222θθγθθεεγθθγθεθεεθθγθεθεε−+−=−+=++=′′′′xyxyyxxyyxyxyyxx(1)These transformation equations involve squares and products of sine and cosine functions andthese can be replaced with double-angle results to yield the double-angle form of thetransformation equations:AE3145 Strain Transformation and Rosette Gage Theory Page 2θγθεεγθγθεεεεεθγθεεεεε2cos2sin)(2sin22cos)()(2sin22cos)()(21212121xyyxyxxyyxyxyxyyxyxx+−−=−−−+=+−++=′′′′(2)Given the ready availability of powerful calculators and spreadsheets, evaluation of thesetransformation equations is a relatively simple matter today and involves little more than a fewseconds to enter the formula in a calculator or spreadsheet cell. This was not always so simple atask and the Mohr’s Circle graphical representation was developed long ago to aid in thisprocess. Today, Mohr’s Circle is not needed for graphical calculation, but it does provide a goodvisualization of the transformation equations and the geometry can be used to infer the actualform for the equations needed for execution in a calculator or computer.The double-angle form of the transformation relations involve simple sine and cosine termsalong with a constant that should suggest equations of a circle with center away from thecoordinate origin. The trick here is to identify the appropriate new “x” and “y” values to plot toconstruct a circle. This is perhaps easier to do by explaining the Mohr’s Circle than it is toactually deduce the form directly. Figure 1 below shows Mohr’s Circle for a state of straindefined by an xy axis. Assume for the moment that the coordinates of the opposite ends (X-Y)of the indicated diameter of the circle define the strain state, εx, εy, and γxy, in the xy axis system.Note that both the εx and εy extensional strains are plotted on the abscissa (x axis) while one-halfthe shear strain, γxy/2, is plotted on the ordinate (y axis). The positive direction for γxy, is takento be downward (consistent with Gere & Timoshenko). One can construct the circle by firstplotting the (εx, γxy/2) pair as point X on the diameter. Next, the pair (εy,-γxy/2) are plotted as theopposite end of the diameter, Y, and the circle can be constructed with X-Y as the diameter.This is the basic Mohr’s Circle and it always has its center on the abscissa at a point given by thevalue, (εx+εy)/2. The circle diameter is easily computed as: D=sqrt[γxy2 + (εx-εy)2].Figure 1. Basic Mohr’s Circle Geometryεx, εyγxy/2γxy/2(εx+εy)/2 (εx-εy)/2 εx ε1 ε2 εyYX0D2φAE3145 Strain Transformation and Rosette Gage Theory Page 3So far, there is no clear connection to the double-angle transformation equations above, butthis will become evident in a moment! To calculate a new state of strain, εx’, εy’ and γx’y’ in anx’y’ axis system rotated θ counterclockwise from the xy axis system, one must construct a newdiameter for Mohr’s Circle rotated 2θ counterclockwise from the initial X-Y diameter as shownbelow in Figure 2. The coordinates of the new diameter endpoints, X’-Y’, represent the newstrain state as computed from the double-angle strain transformation equations. This should beclear by inspection of the Mohr’s Circle and by evaluation of the circle geometry as shown.Note that rotation in Mohr’s Circle is always 2x the geometric rotation so that the state of straindefined by opposite ends of a diameter of the circle (e.g., 180° apart) correspond to strains thatare at 180°/2=90° in geometric axes.Figure 2. Mohr’s Circle for Strain Transformation of θIt should be pointed out also that this particular form for Mohr’s Circle uses an invertedordinate (y axis) and positive θ is counterclockwise. Another popular form for Mohr’s Circleuses an upward positive ordinate