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Rosette Strain Gauges

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Rosette Strain Gauges

 

A single strain gauge can only measure strain in one direction.  In real life applications, this is often inadequate due to the complex nature of most structures and their loads.

 

Strains and stresses may come in various directions and thus a gauge capable of measuring several different directions simultaneously is necessary.

 

The Challenges with Single Strain Gauges

 

In Figure 1, would the single, uniaxial gauge capture the strain field correctly?

 

Only the strain gauge on the left properly measures the strain field.  A single uniaxial strain gauge only measures the strain field correctly in one direction. To measure more complicated strain fields, a rosette strain gauge may be required.

 

Figure 1: Three different force loads on a metal cylinder. Only for the axial load case on the left does the uniaxial strain gauge measure properly. For the torsional and bending loads on the right, the uniaxial strain gauge is insufficient.Figure 1: Three different force loads on a metal cylinder. Only for the axial load case on the left does the uniaxial strain gauge measure properly. For the torsional and bending loads on the right, the uniaxial strain gauge is insufficient.

Most real systems/products have complicated geometries and multi-directional loads that cannot be measured by an individual strain gauge.

 

Instead of thinking of the strain in a single, uniaxial direction, a planar approach can be used to think of strain in a XY axis system as shown in Figure 2.  

 

Figure 2: For complicated strain fields, a planar approach can be used.Figure 2: For complicated strain fields, a planar approach can be used.

 

In a plane, strain can manifest itself in three ways:

 

  • Normal strain in X direction  (ɛx)
  • Normal strain in Y direction (ɛy)
  • Shear strain in XY ( ɛxy)

 

Strain Tensor vs. Principal Strain

 

There are two methods to define strain in a plane: strain tensor or principal strain.  Both methods define the same planar strain state at a point on a test piece, but with a different “perspective”:

 

Method 1: Strain Tensor - The first method considers three strain components: two normal components (ɛx,ɛy) and a shear component called γxyor ɛxy. The strains are considered in the xy coordinate system as shown in Figure 2 (left side).

 

Method 2: Principal Strain - Two principal strains and an angle are used.  The gauge is “virtually” rotated so that the shear strain is zero, leaving the two largest principal strain components in the plane. The angle of the principal strain indicates how it is rotated relative to the XY axis as shown in Figure 3 (right side).

 

 

Figure 3: Left is Strain Tensor, Right is Principal StrainFigure 3: Left is Strain Tensor, Right is Principal Strain

Correctly identifying the principal (ie, largest) strain is very important.The fatigue life of the part is determined by the largest strain.  If a smaller component strain was used in a fatigue life calculation, it would be under-estimated and the part would fail sooner than predicted.

 

The two principal strains and angle are related to the strain tensor by a series of equations known as the “strain-transformation.”

 

The “strain-transformation” can be easily visualized with the aid of Mohr’s circle (Figure 4). Mohr’s circle plots the normal strain (x axis) with respect to the shear strain (y axis) and provides a model by which both the principal strain and the maximum shear can be determined.

 

Figure 4: Mohr's circleFigure 4: Mohr's circle

The Mohr’s circle has the following properties:

  • A strain tensor consisting of two normal (ɛx, ɛy) strains and a shear strain (ɛxy) are calculated from the measured rosette strain gauge arms ɛ1, ɛ2, and ɛ3 as shown in Figure 6 or Figure 7
  • The average strain is calculated as (ɛx+ ɛy)/2 and plotted on the X axis where the shear strain is zero.
  • Using the calculated normal and shear strains, two points (ɛx, ɛxy) and (ɛy, -ɛxy) are plotted on the graph. 
  • A circle is fit thru the two points (ɛx, ɛxy) and (ɛy, -ɛxy) with the average strain as the center of the circle.
  • The two principal strains are the minimum and maximum values where the Mohr’s circle intersects the x axis at zero shear strain.
  • The angles of the Mohr's circle are twice the angles of the rosette gauge. Depending on the convention used, the angle will either be between 0 and 180 degrees, or between -90 and +90 degrees.
  • Note that Mohr’s circle is constructed with positive shear strain plotted downward. This is done so that the positive rotational direction of the angle in Mohr’s circle is the same (CCW) as for the rosette.

Commonly, the normal strain and the shear strain output of a CAE simulation is based on the strain tensor method. The strain output of a test using a rosette gauge is based on the principal strain method. To compare the output of a CAE simulation to a test, the “strain-transformation” must be used.

 

Rosette Strain Gauge Calculations

 

A rosette strain gauge can be used to capture multi-directional strain fields and determine the principal (ie, largest) strains at any given location on a test piece at any point in time.

 

Strain gauge rosettes combine three co-located strain gauges at specific fixed angles to measure the normal strains along the surface of a test part as shown in Figure 5.

 

Figure 5: Rosette Strain Gauge consists of three co-located strain gauges.Figure 5: Rosette Strain Gauge consists of three co-located strain gauges.

In theory, each strain gauge should measure the strain at the same position on the part. This is done by placing the gauges in a tight grouping near the rosette center. Rosette gauges even come in "stacked" configurations if strain needs to be measured on the exact same point due to large strain gradients.

 

The three strain gauge measurements, Young’s modulus of the material, and Poisson’s Ratio are used to calculate the following nine different values from a rosette strain gauge:

 

  • principal strain 1 (SN1) : the strain in the direction of the principal stress 1
  • principal strain 2 (SN2) : the strain in the direction of the principal stress 2
  • shear strain (SNSH)
  • angle (AG) : the angle from strain 1 to the principal axis
  • principal stress 1 (SS1) : the maximum principal stress
  • principal stress 2 (SS2) : the minimum principal stress
  • shear stress (SNSH)
  • equivalent stress (ES) : the equivalent stress according to von Mises
  • biaxiality ratio (BR) : the ratio of the two principal stresses

 

Three actual measurements give nine calculated outputs!  That’s a three to one return!

 

Delta and Rectangular Rosettes

 

Rosette strain gauges have two common configurations: rectangular or delta. These configurations simplify much of the math involved in the rosette calculations.

 

Rectangular Rosettes

 

Rectangular Rosettes separate gauges by 45° placing a strain gauge on both the X and Y coordinate axes as seen in Figure 6.

 

Due to the placement of the gauges, the math for a rectangular gauge is more simple than a delta gauge.  With today’s computers, this is not an important criteria to consider when selecting rosette gauges. 

 

Figure 6: Rectangular Rosette diagramFigure 6: Rectangular Rosette diagramThe following formulas are used to calculate the nine outputs of a rectangular rosette gauge:

 

RectRosetteEquations.png

 

Delta Rosette Gauges

 

Delta gauges have a wider coverage versus rectangular gauges.  The strain gauges are separated by 60°, and the middle strain gauge is aligned with the y-axis as shown in Figure 7.

 

Figure 7: Delta Rosette diagramFigure 7: Delta Rosette diagram

The following formulas are used to calculate the nine outputs of a delta rosette gauge:

 

DeltaRosetteEquations.png

 

Biaxiality Ratio

 

When doing the calculations for a rosette gauge, a biaxiality ratio can also be calculated. The biaxiality ratio is the ratio of the two principal stresses (SS1 and SS2) as seen in Equation 1 (assuming |SS1| > |SS2|). The principal stress with the largest absolute value is always put in the denominator so that the biaxiality values are always between -1 and 1.

 

Equation 1: Biaxiality Ratio is the ratio of Principal StressesEquation 1: Biaxiality Ratio is the ratio of Principal Stresses

The biaxiality ratio can be any value between -1 and 1:

  • 0 : If the biaxiality ratio is 0, the stress/strain field is uniaxial tension or compression
  • -1 : If the biaxiality ratio is -1, the stress/strain field is pure shear stress or strain
  • 1 : If the biaxiality ratio is 1, the stress/strain is equal in all directions

The biaxiality ratio is one of the parameters calculated using the ROSETTE virtual channel calculations in LMS Test.Lab.

 

Rosette Gauges in LMS Test.Lab

 

Rosette strain gauges can be setup via “Virtual Channels” in LMS Test.Lab. In Signature acquisition, change “Channel setup” to “Virtual Channels” using the pulldown in the upper right of the “Channel Setup” worksheet as shown in Figure 8.

 

 

Figure 8: Virtual Channels in LMS Test.Lab Channel Setup worksheet.Figure 8: Virtual Channels in LMS Test.Lab Channel Setup worksheet.

After selecting “Virtual Channels” a formula area appears at the bottom of the Channel Setup worksheet as shown in Figure 9. 

 

Figure 9: Function Selection in Virtual Channels via the "f(x)" buttonFigure 9: Function Selection in Virtual Channels via the "f(x)" button

Click on the “insert function” button with the “f(x)” symbol and select “Strain gauges” group of functions. Then, select the type of strain gauge that is being used in the test: delta or rectangular.

 

In the “Edit formula arguments” menu, enter the three channels of the Rosette strain gauge, Young’s Modulus and Poisson’s ratio (Figure 10).

 

Figure 9: “Edit formula arguments” menu for Rosette gaugeFigure 9: “Edit formula arguments” menu for Rosette gauge

Note: Young’s modulus is 210000 MPa for a typical steel.

 

Press the ‘OK’ button on the ‘Edit formula arguments’ menu when finished.  Nine new rosette time calculation channels will be created in the resulting time history file as seen in Figure 11.

 

Figure 11: Rosette Time Calculation channels in Time FileFigure 11: Rosette Time Calculation channels in Time File

Rosette strain gauge calculations can also be performed offline using the LMS Test.Lab Time Signal Calculator.

 

Enjoy measuring rosette strain gauges! Further questions? Contact Us!

 

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