Study and analysis of strain rosette

1. STRAIN ROSETTE
ANALYSIS
By,
TUSHAR GHAG
ME-MACHINE DESIGN
ROLL NO-221603

2. Introduction
A strain gauge is a passive transducer that converts mechanical displacement
into change of resistance. It is a thin, wafer like material than can be attached
to a variety of material to measure their strain.
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3. Principle: The basic concept of an electrical strain gauge is attributed to Lord
Kelvin who in 1856 expounded the theory that the resistance of a copper or
iron wire changes when subjected to tension. The resistance of the wire
changes as a function of strain, increasing with tension and reducing with
compression.
𝑅 =
𝜌𝐿
𝐴
Types Of Electrical Strain Gauges:
1) Wire Gauges
2) Wrap-around Type
3) Unbonded Strain Gauges
4) Foil Gauges
5) Bonded Strain Gauge
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4. Axial Sensitivity
• It is very important that the strain gauge be properly mounted onto the test
specimen so that the strain is accurately transferred from the test specimen,
through the adhesive and strain gauge backing, to the foil itself.
• A fundamental parameter of the strain gauge is its sensitivity to strain,
expressed quantitatively as the Gauge Factor (GF).
• Gauge factor is defined as the ratio of fractional change in electrical
resistance to the fractional change in length (strain). The gauge factor for
metallic strain gauges is typically around 2.
𝐺 =
∆𝑅/𝑅
∆𝐿/𝐿
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5. Cross Sensitivity
• Transverse sensitivity in a strain gauge refers to the behaviour 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.
• However 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.
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6. Cross Sensitivity
• In simple words, the gauge in addition to measuring the strain along the
axis also measures the strain transverse to it. This is known as cross-
sensitivity or transverse sensitivity.
Axial Strain Sensitivity:
𝑆|| =
∆𝑅/𝑅
∈𝑥
Normal Strain Sensitivity:
𝑆𝑁 =
∆𝑅/𝑅
∈𝑦
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7. ➢ CROSS SENSITIVITY FACTOR IS GIVEN BY,
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8. Strain Gauge Rosettes
Introduction:
• 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.
• It is often desired to measure the full state of strain on the surface of a part,
i.e. to measure not only the two extensional strains, 𝜀𝑥 and 𝜀𝑦 but also the
shear strain 𝛾𝑥𝑦 with respect to some given xy-axis system.
• A single gauge is capable only of measuring the extensional strain in the
direction that the gauge is oriented.
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9. • Rosettes are designed to perform a very practical and important function in
experimental stress analysis.
• It can be shown that for a 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.
• When the principal directions are known in advance, two independent
strain measurements are needed to obtain the principal strains and stresses.
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10. • To meet the requirements, manufacturers offer three basic types of strain
gauge rosettes (each in a variety of forms):
 Tee: Two mutually perpendicular grids.
 45°- rectangular: Three grids, with the second and third grids angularly
displaced from the first grid by 45° and 90°, respectively.
 60°- delta: Three grids, with the second and third grids 60° and 120°
away, respectively, from the first grid.
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11. Representative gauge patterns for the three rosette types are reproduced in
figure.
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12. Rosette Selection Considerations
• Basic parameters are strain-sensitive alloy, backing material, self
temperature-compensation number, gauge length which must be considered
in the selection of any strain gauge. Two other parameters are important in
rosette selection. These are: (1) the rosette type — tee, rectangular or delta
and (2) the rosette construction — planar or stacked .
• The tee rosette should be used only when the principal strain directions are
known in advance from other considerations.
• If there is uncertainty about the principal directions, a three-element
rectangular or delta rosette is preferable.
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13. • All three types of rosettes (tee, rectangular, and delta) are manufactured in both
planar and stacked versions.
• The stacked rosette will cover more area than planar rosette and hence will give
an accurate result.
• However if this rosette is mounted on a thin member subjected to severe
bending, a considerable error will be introduced since each gauge is at different
distance from neutral axis.
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14. Gauge Element Numbering
• “Numbering” refers to the numeric sequence in which the gauge elements
in a rosette are identified during strain measurement.
• With any three-element rosette, misinterpretation of the rotational sequence
(cw or ccw) can lead to incorrect principal strain directions
• To obtain correct results, the grids in three-element rosettes must be
numbered in a particular way.
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15. Gauge Element Numbering
• It is always necessary in a tee rosette, for instance, that grid numbers 1 and
3 be assigned to two mutually perpendicular grids. Any other arrangement
will produce incorrect principal strains.
• Following are the general rules for proper rosette numbering:
 With a rectangular rosette, the axis of grid 2 must be 45° away from
that of grid 1; and grid 3 must be 90° away, in the same rotational
direction.
 With a delta rosette, the axes of grids 2 and 3 must be 60° and 120°
away, respectively, in the same direction from grid 1.
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16. • Numbering the grids can be in either the clockwise or counterclockwise
direction, as long as the sequence is correct. Counterclockwise numbering
is preferable, because it is consistent with the usual engineering practice of
denoting counterclockwise angular measurement as positive in sign.
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17. Stress Strain Relationship
𝜀𝑥′ =
𝜀𝑥+𝜀𝑦
2
+
𝜀𝑥−𝜀𝑦
2
𝑐𝑜𝑠 2𝜃 +
𝛾𝑥𝑦
2
𝑠𝑖𝑛 2𝜃
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18. Stress Strain Relationship
• 𝜀𝑎 =
𝜀𝑥+𝜀𝑦
2
+
𝜀𝑥−𝜀𝑦
2
cos 2𝜃𝑎 +
𝛾𝑥𝑦
2
sin 2𝜃𝑎
• 𝜀𝑏 =
𝜀𝑥+𝜀𝑦
2
+
𝜀𝑥−𝜀𝑦
2
cos 2𝜃𝑏 +
𝛾𝑥𝑦
2
sin 2𝜃𝑏
• 𝜀𝑐 =
𝜀𝑥+𝜀𝑦
2
+
𝜀𝑥−𝜀𝑦
2
cos 2𝜃𝑐 +
𝛾𝑥𝑦
2
sin 2𝜃𝑐
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19. Stress Strain Relationship
Having three unknowns and three equations, 𝜀𝑥 , 𝜀𝑦 and 𝛾𝑥𝑦 can be evaluated.
The principal strains are then given by;
1. 𝜀𝐼 =
𝜀𝑥+𝜀𝑦
2
+
𝜀𝑥−𝜀𝑦
2
2
+ 𝛾𝑥𝑦
2
2. 𝜀𝐼𝐼 =
𝜀𝑥+𝜀𝑦
2
−
𝜀𝑥−𝜀𝑦
2
2
+ 𝛾𝑥𝑦
2
3. 𝑡𝑎𝑛 2𝜃 =
𝛾𝑥𝑦
𝜀𝑥−𝜀𝑦
4. 𝛾𝑚𝑎𝑥 = (𝜀𝑥 − 𝜀𝑦)2+ 𝛾𝑥𝑦
2
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20. Strain Rosette - 45°
• To increase the accuracy of a strain rosette,
large angles are used.
• In a three gauge rectangular rosette gauges
are separated by 45o; 𝜃𝑎 = 0°, 𝜃𝑏 = 45° and
𝜃𝑐 = 90°. The three equations can then be
simplify to:
1. 𝜀𝑎 =
𝜀𝑥+𝜀𝑦
2
+
𝜀𝑥−𝜀𝑦
2
2. 𝜀𝑏 =
𝜀𝑥+𝜀𝑦
2
+
𝛾𝑥𝑦
2
3. 𝜀𝑐 =
𝜀𝑥+𝜀𝑦
2
+
𝜀𝑥−𝜀𝑦
2
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21. Strain Rosette - 45°
Solving for 𝜀𝑥, 𝜀𝑦 and 𝛾𝑥𝑦 gives,
1. 𝜀𝑥 = 𝜀𝑎
2. 𝜀𝑦 = 𝜀𝑐
3. 𝛾𝑥𝑦 = 2𝜀𝑏 − (𝜀𝑎 + 𝜀𝑐)
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22. STRAIN ROSETTE - 60O
• Similarly, if the angles between the
gauges are , 60o; 𝜃𝑎 = 0°, 𝜃𝑏 = 60° and
𝜃𝑐 = 120°. The unknown strains, for 𝜀𝑥 ,
𝜀𝑦 and 𝛾𝑥𝑦 will be,
1. 𝜀𝑥 = 𝜀𝑎
2. 𝜀𝑦 =
2𝜀𝑎+2𝜀𝑐−𝜀𝑎
3
3. 𝛾𝑥𝑦 =
2𝜀𝑏−2𝜀𝑐
3
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23. Applications
1. Structural health monitoring ( railroads, bridges, dams etc.)
2. Vehicle classifications using strain gauge
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