Strain Rosettes
STRAIN ROSETTES DEFINATION A strain gauge rosette is, by definition, an arrangement of two or more closely positioned gauge grids, separately oriented to measure the normal strains along different directions in the underlying surface of the test part.
INTRODUCTION Rosettes are designed to perform a very practical and important function in experimental stress analysis. It can be shown that for the not-uncommon case of the general biaxial stress state, with the principal directions unknown, three independent strain measurements (in different directions) are required to determine the principal strains and stresses. And even when the principal directions are known in advance, two independent strain measurements are needed to obtain the principal strains and stresses. In common with single-element strain gauges, rosettes are manufactured from different combinations of grid alloy and backing material to meet varying application requirements. They are also offered in a number of gauge lengths, noting that the gauge length specified for a rosette refers to the active length of each individual grid within the rosette. To meet the foregoing requirements, the Micro-Measurements Division manufactures three basic types of strain gauge rosettes (each in a variety of forms): 1. Tee Tee (0(0-90 90 degr degree ee)) Two mutually perpendicular grids.
Tee 2. Rectan Rectangul gular ar (0-4 (0-455-90 90 degr degree ee)) Three grids, with the second and third grids angularly displaced from the first grid by 45 degrees and 90 degrees, respectively.
Rectangular rosette 1
Strain Rosettes
3. Delt Delta a (0-60 (0-60-1 -120 20 degr degree ee)) Three grids, with the second and third grids 60 degrees and 120 degrees away, respectively, from the first grid.
Delta rosette 4. Stacked Co-location of the gauges requires mounting each individual gauge on top of the others in what is called a “stacked” rosette, but this leads to a complicated and often inaccurate type of gauge.
Stacked rosette
Rosette selection consideration The initial step in preparing for any strain gauge installation is the selection of the appropriate gauge for the task. It might at first appear that gauge selection is a simple exercise, of no great consequence to the stress analyst; but quite the opposite is true. Careful, rational selection of gauge characteristics and parameters can be very important in: optimizing the gauge performance for specified environmental and operating conditions, obtaining accurate and reliable strain measurements, contributing to the ease of installation, and minimizing the total cost of the gauge installation. The installation and operating characteristics of a strain gauge are affected by the following parameters, which are selectable in varying degrees: • Strain-sensitive alloy • Backing materials (carrier) • Grid resistance 2
Strain Rosettes • Gauge pattern • Self-temperature compensation number • Gauge length • Options Basically, the gauge selection process consists of determining the particular available combination of parameters which is most compatible with the environmental and other operating conditions , and at the same time best satisfies the installation and operating constraints . These constraints are generally expressed in the form of requirements such as: • Accuracy • test duration • Stability • Cyclic endurance • Temperature • ease of installation • Elongation • Environment The cost of the strain gauge itself is not ordinarily a prime consideration in gauge selection, since the significant economic measure is the total cost of the complete installation, of which the gauge cost is usually but a small fraction. In many cases, the selection of a gauge series or optional feature which increases the gauge cost serves to decrease the total installation cost. It must be appreciated that the process of gauge selection generally involves compromises. This is because parameter choices which tend to satisfy one of the constraints or requirements may work against satisfying others. For example, in the case of a smallradius fillet, where the space available for gauge installation is very limited, and the strain gradient extremely high, one of the shortest available gauges might be the obvious choice. At the same time, however, gauges shorter than about 0.125 in [3 mm] are generally characterized by lower maximum elongation, reduced fatigue life, less stable behavior, and greater installation difficulty.
Two-Element Rosettes The tee rosette should be used only when the principal strain directions are known in advance from other considerations. Attention must also be given to nearby geometric irregularities, such as holes, ribs, or shoulders, which can locally alter the principal directions. When necessary, the tee rosette can be installed with its axes at any precisely known angle from the principal axes; but greatest accuracy will be achieved by alignment along the principal directions.
Three-Element Rosettes Where the directions of the principal strains are unknown, a three-element rectangular or delta rosette is always required; and the rosette can be installed without regard to orientation. Because the gauge axes in the delta rosette have the maximum possible uniform angular separation (effectively 120 degrees), this rosette is presumed to produce the optimum sampling of the underlying strain distribution. The choice between rectangular and delta rosettes is more apt to be based on practical application considerations such as availability
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Strain Rosettes from stock, compatibility with the space available for installation, convenience of solder tab arrangement, etc.
Construction All three types of rosettes (tee, rectangular, and delta) are manufactured in both planar and stacked versions. As indicated (for the rectangular rosette) below, the planar rosette is etched from the strain-sensitive foil as an entity, with all gauge elements lying in a single plane. The stacked rosette is manufactured by assembling and laminating two or three properly oriented single-element gauges
When strain gradients in the plane of the test part surface are not too severe, the normal selection is the planar rosette. This form of rosette offers the following advantages in such cases: • Thin and flexible, with greater conformability to curved surfaces. • Minimal reinforcing effect. • Superior heat dissipation to the test part. • Available in all standard forms of gauge construction, and generally. • Accepts all standard optional features. • Optimal stability. Maximum freedom in lead wire routing and attachment. • The principal disadvantages of the planar rosette arise from the larger surface area covered by the sensitive portion of the gauge. When the space available for gauge installation is small, a stacked rosette may fit, although a planar one will not. More importantly, where a steep strain gradient exists in the surface plane of the test part, the individual gauge elements in a planar rosette may sense different strain fields and magnitudes. For a given active gauge length, the stacked rosette occupies the least possible area, and has the centroids (geometric centers) of all grids lying over the same point on the test part surface. Thus, the stacked rosette more nearly approaches measurement of the strains at a point. Although normally a trivial consideration, it can also be noted that all gauges in a stacked rosette have the same gauge factor and transverse sensitivity, while the grids in a planar rosette will differ slightly in these properties, due to their different orientations relative to the rolling direction of the
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Strain Rosettes strain-sensitive foil. The technical data sheet accompanying the rosettes fully documents the separate properties of the individual grids. It should be realized, however, that the stacked rosette is noticeably stiffer and less conformable than its planar counterpart. Also, because the heat conduction paths for the upper grids in a stacked rosette are much longer, the heat dissipation problem may be more critical when the rosette is installed on a material with low thermal conductivity. Taking into account their poorer heat dissipation and their greater reinforcement effects, stacked rosettes may not be the best choice for use on plastics and other nonmetallic materials. A stacked rosette can also give erroneous strain indications when applied to a thin specimen in bending, since the grid plane of the uppermost gauge in a three-gauge stack may be as much as 0.0045 in (0.11 mm) above the specimen surface. In short, the stacked rosette should ordinarily be reserved for applications in which the requirement for minimum surface area dictates its selection.
Principal Strains & Direction From Measurements 5
Strain Rosettes
The equations for calculating principal strains from three rosette strain measurements are derived from what is known as a "strain-transformation" relationship. This situation can be envisioned most readily with the aid of the well-known Mohr's circle for strain. It can be seen from this figure that the normal strain at any angle the major principal axis is simply expressed by:
from
Thus for a rectangular rosette the situation will be following:
Rectangular rosette oriented on an element
Mohr’s circle for rect rosette
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Strain Rosettes Hence the equations for this case will be:
And hence the principal strains calculated from this case will be
And in the case of delta rosette the same procedure can be used to derive corresponding data-re data-redu ductio ction n equatio equations ns for for the delt deltaa rosette rosette shown shown belo below. w. When When grid grid angles angles , +60 and +120 the resulting three equations can again be solved simultaneously for the principal strains and angle. Thus, for the delta rosette:
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Strain Rosettes
Transverse Sensitivity Transverse sensitivity in a strain gauge refers to the behavior of the gauge in responding to strains which are perpendicular to the primary sensing axis of the gauge. Ideally, it would be preferable if strain gauges were completely insensitive to transverse strains. In practice, most gauges exhibit some degree of transverse sensitivity; but the effect is ordinarily quite small, and of the order of several percent of the axial sensitivity. In plane wire strain gauges, transmission of strain into the wire from a direction perpendicular to the wire axis is nearly negligible. As a result, the transverse sensitivity of these gauges is due almost entirely to the fact that a portion of the wire in the end loop lies in the transverse direction. Because of this, the sign of the transverse sensitivity for a plane wire gauge will always be positive, and the magnitude of the effect can be calculated quite closely from the geometry of the grid. This statement does not apply to the small "wrap-around" gauges having the wire wound on a flattened core. Such gauges often exhibit negative transverse sensitivities. In foil strain gauges, on the other hand, the transverse sensitivity arises from much more complex phenomena, and it is affected by almost every aspect of grid design and gauge construction. In addition to end-loop effects, the foil grid lines, having a large ratio of width to thickness, are strained significantly by transverse strains. The magnitude of transverse strain transmission into the grid lines is determined by the relative thicknesses and elastic moduli of the backing and foil, by the width-to-thickness ratio of the foil grid lines, and, to a lesser degree, by several other parameters, including the presence or lack of an encapsulating layer over the grid. Depending upon the foil material and its metallurgical condition, the contribution to transverse sensitivity from the transmission of transverse strain into the grid lines can be 8
Strain Rosettes either positive or negative. Because of this, the overall transverse sensitivity of a foil strain gauge can also be either positive or negative. While the transverse sensitivity of a foil gauge is thus subject to a greater degree of control in the design of the gauge, the compromises necessary to optimize all aspects of gauge performance generally limit the attainable reduction in transverse sensitivity.
Errors, Corrections & Limitations The obvious aim of experimental stress analysis is to determine the significant stresses in a test object as accurately as necessary to assure product reliability under expected service conditions. As demonstrated in the preceding sections of this publication, the process of obtaining the principal stresses involves three basic, and sequential, steps: 1. Measurement of surface strains with a strain gauge rosette. 2. Transforma Transformation tion of of measured measured strains strains to princip principal al strains. strains. 3. Conversion of principal strains to principal stresses.
Each step in this procedure has its own characteristic error sources and limits of applicability; and the stress analyst must carefully consider these to avoid potentially serious errors in the resulting principal stresses. Of first importance is that the measured strains be as free as possible of error. Strain measurements with rosettes are subject, of course, to the same errors (thermal output, transverse sensitivity, lead wire resistance effects, etc.) as those with single-element strain gauges. Thus, the same controlling and/or corrective measures are required to obtain accurate data.
I. Transve Transverse rse Sen Sensit sitivi ivity ty Errors in strain indication due to transverse sensitivity are generally quite small since the transverse sensitivity itself is small. However, in biaxial strain fields characterized by extreme ratios between principal strains, the percentage error in the smaller strain can be very great if not corrected for transverse sensitivity. At least one of the gauge grids in any rosette will in every case be subjected to a transverse strain which is equal to or greater than the 9
Strain Rosettes strain along the grid axis, consideration should always be given to the transverse-sensitivity error when performing rosette data reduction. The magnitude of the error in any particular case depends on the transverse-sensitivity coefficient ( K K t t ) of the gauge grid, and on the ratio of the principal strains. In general, when K t t < 1%, the transverse-sensitivity error is small enough to be ignored. However, at larger values of K t t , depending on the required measurement accuracy, correction for transverse sensitivity may be necessary.
Correction for transverse sensitivity error The effects of transverse sensitivity should always be considered in the experimental stress analysis of a biaxial stress field with strain gauges. Either it should be demonstrated that the effect of transverse sensitivity is negligible and can be ignored, or, if not negligible, the proper correction should be made. Since a two- or three-gauge rosette will ordinarily be used in such cases, simple correction methods are given here for the two-gauge 90-degree rosette, the three-gauge rectangular rosette, and the delta rosette. Consider first the two-gauge 90-degree rosette, with the gauge axes aligned with two orthogonal axes, x and y, on the test surface. When using this type of rosette, the x and y axes would ordinarily be the principal axes, but this need not necessarily be so. The correct strains along any two perpendicular axes can always be calculated from the following equations in terms of the indicated strains along those axes:
Where:
respectively. The (1– Kt ) term in the denominators of Equations is generally in excess of 0.995, and can be taken as unity : 2
II.Thermal Effects When strain measurements must be made in a variable thermal environment, the thermal output of the strain gauge can produce rather large errors, unless the instrumentation can be zero-balanced at the testing temperature, under strain-free conditions. In addition, the gauge factor of the strain gauge changes slightly with temperature.
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Strain Rosettes
III.Grid Alignment After making certain that strain measurement errors such as the foregoing have been eliminated or controlled to the degree feasible, attention can next be given to possible errors in the strain-transformation procedure for obtaining the principal strains. A potentially serious source of error can be created when the user attempts to make up a rosette on the specimen from three conventional single-element gauges. The error is caused by misalignment of the individual gauges within the rosette. If, for example, the second and third gauges in a rectangular rosette configuration are not accurately oriented at 45 degrees and 90 degrees, respectively, from the first gauge, the calculated principal strains will be in error. The magnitude of the error depends 1. Dire Directi ction on of of misal misalig ignm nmen ent. t. 2. Rati Ratio o of pri princ ncip ipal al strai strains ns.. 3. Overall Overall orientatio orientation n of rosette rosette with respect to princip principal al axis. axis. These quantities are defined in Figures 1 for the particular but common case of the uniaxial stress field. Figure 1 is a polar diagram of strain at the point in question, and Figure 2 gives the concentric Mohr’s circles for stress and strain for the same point. In Figure 1, the distance to the boundary of the diagram along any radial line is proportional to the normal strain along the same line. The small lobes along the Y axis in the diagram represent the negative Poisson strain for this case. It can be seen qualitatively from Figure 1 that when φ is 0 or 90, a small angular misalignment of the gage will produce a very small error in the strain indication, since the polar strain diagram is relatively flat and passing through zero slope at these points.
FIGURE 1
It would be interesting to know that for certain combinations of principal strain ratio and rosette orientation, 5-degree alignment errors in Gauges 2 and 3 relative to Gauge 1 can produce an error of 20 percent or more in one of the principal strains. This shows the significance of the misalignment error and illustrates that problems that occur due to it can be very great due to their magnitude. The error in strain indication due to angular misalignment of the gage can be expressed as follows: n = ε ( φ∓β) -ε φ 11
Strain Rosettes Where: n = Error, με φ = Strain along axis of intended measurement at angle φ from principal axis, με ε ( φ∓β ) = Strain along gage axis with angular mounting error of ± β, με
Since it is very difficult for most persons to install a small strain gauge with the required precision in alignment, the user is well-advised to employ commercially available rosettes. The manufacturing procedures for Micro-Measurements strain gauge rosettes are such that errors due to grid alignment within the rosette need never be considered. For those cases in which it is necessary, for whatever reason, to assemble a rosette from single-element gauges, extreme care should be exercised to obtain accurate gauge alignment.
I.
Uniform Strain Field Under Gauge
The strain-transformation relationships and data-reduction equations given herein assume a uniform state of strain at the site of the rosette installation. Since the rosette necessarily covers a finite area of the test surface, severe variations in the strain field over this area can produce significant errors in the principal strains - particularly with planar rosettes. For this type of application, the stacked rosette is distinctly superior; both because it covers a much smaller area (for the same gauge length), and because the centroids of all three grids lie over the same point on the test surface . A case in point is the use of strain gauge rosettes on fiber-reinforced composite materials. If the distance between inhomogeneities in the material (i.e., fiber-to-fiber spacing) is small compared to the gauge length of the rosette, each grid will indicate the "macroscopic" or average strain in the direction of its axis.
II.Large Strains There is an additional limitation to the strain-transformation relationship in which, although not frequently encountered in routine experimental stress analysis, should be noted. The subject of the strain distribution about a point, as universally treated in handbooks and in mechanics of materials textbooks, is developed from what is known as "infinitesimal-strain" theory. That is, in the process of deriving relatively simple relationships the strain magnitudes are assumed to be small enough so that normal- and shear-strain approximations of the following types can be employed without introducing excessive error:
Although often unrecognized, these approximations are embodied in the equations used throughout the contemporary practice of theoretical and experimental stress analysis (where strain transformation is involved). This includes the concept of Mohr's circle for strain, and thus all of the preceding equations, which are consistent with the strain circle. Infinitesimal-strain theory has proven highly satisfactory for most stress analysis applications with conventional structural materials, since the strains, if not truly "infinitesimal", are 12
Strain Rosettes normally very small compared to unity. Thus, for a not-untypical working strain level of 0.002 (2000 micro strain), the error in ignoring to is only about 0.2 percent. However, strain gauge rosettes are sometimes used in the measurement of much larger strains, as in applications on plastics and elastomers, and in post-yield studies of metal behavior. Strain magnitudes greater than about 0.01 (10 000 micro strain) are commonly referred to as "large" or "finite"; and, for these, the strain-transformation relationship may not adequately represent the actual variation in strain about a point. Depending on the strain magnitudes involved in a particular application, and on the required accuracy for the principal strains, it may be necessary to employ large-strain analysis methods for rosette data reductions.
Reason Why 60o Rosette And 120o Rosette Have Same Expressions When a comparison is made between the strain equations of 60 o rosette and 120o rosette it becomes clear that there is not a great difference among these expressions. Actually the equations for the normal strains are same and when it comes to shear strain γ xy (for 120o strain rosettes) = −γ xy (for 60o strain rosettes)
A Comparison Among 60 o And 120o Strain Rosettes
not
The reason is that when in strain formulae its 2θ 2 θ which is taken into the account . Focusing on the above figures it becomes clear that value of
ε
b
(for 120o) = ε
c
(for 60o) and ε b(for 60o) =ε =ε
c
(for 120o)
Because 2×240=480= 360+120 and after one complete c omplete revolution only 120 o will be taken in account which is equal to 2 for rosette b of 60 o rosette.
So now ε b + ε c gives the value of ε of ε y but as it is now clear that value of b gauge for o 120 is value of c gauge for 60 and vice versa but since it is the sum which gives value of ε so there will be no change in expression for ε for ε y but while calculating shear strain we know o
y
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Strain Rosettes that its ε b-ε c in case of 60 o rosette but since values of ε of ε b ε c are interchanged for 120o rosette so it will no longer b ε b -ε c but –ve of it i.e. – ( ε b - ε c) This the reason why the normal strains for 60 o and 120 o rosettes have same value but shear strains are –ve of each other.
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