Lecture#8
- Strain Gages
- Electrical Resistance Strain Gages
- Strain Gage Rosettes
- transformation matrix
Dr.Haydar Al-Ethari
References:
1- James Doyle, 2004, Modern Experimental Stress Analysis: completing
the solution of partially specified problems. John Wiley & Sons, Ltd.
2- Dieter G.E., (1986), Mechanical Metallurgy, McGraw-Hill.
3- Hearn E. J., (2000), Mechanics of Materials, vol. 1&2., Butterworth &
Heinemann.
Strain Gages:
A transducer is a device that transforms one type of energy into another.
These are of great importance in measurement because, generally, it happens
that the original quantity (strain, say) to be measured is too small to handle
in its original form and instead an intermediate quantity is used. In modern
measurements, the most common intermediate quantity is the electric
voltage.
Electrical Resistance Strain Gages:
Probably the most reliable of all the tools of experimental stress analysis is
the electrical resistance strain gauge. The principle of its action is that the
electrical resistance of a conductor changes proportionally to any strain
applied to it. Thus, if a short length of wire were bonded to the structure in
such a way that it experiences the same deformation as the structure, then by
measuring the change in resistance, the strain can be obtained.
Basic Theory
Consider a conducting wire of length L and area A as shown. Its resistance
(neglecting temperature effects) is proportional to its length but inversely
proportional to the area, that is,
R = L/A
where is a constant of proportionality, called the specific resistivity and is
essentially inversely proportional to the number of mobile electrons per unit
volume (N/V).
1
where K is a proportionality factor, the resistance is written as:
Suppose this wire is stretched; there will be a small change in resistance
given by:
or
The number of mobile electrons is generally proportional to stress (and
hence strain = L/L), that is:
and
It follows that any change in length, and hence sectional area, will result in a
change of resistance. Thus measurement of this resistance change enables a
direct reading of linear strain to be obtained. This is made possible by the
relationship which exists for a number of alloys over a considerable strain
range between change of resistance and strain which may be expressed as
follows:
∆R
∆L
= Sm
R
L
where R and L are the changes in resistance and length respectively and
Sm is termed the material sensitivity or gauge factor.
Typical values of Sm for most conventional gauges lie in the region of 2 to
2.2, and most strain-gauge instruments allow the value of Sm to be set
accordingly, thus enabling strain values to be recorded directly.
An actual gauge is not a straight length of wire but is in the form of a grid as
shown in the figure. The original wire gauges were wound back and forth
over a paper former so as to maintain a large resistance but small size.
2
The gauge sensitivity or gauge factor is measured in a uniaxial stress field
and this will give rise to an error when the gauge is used in other than the
calibrated situation. This is known as the transverse sensitivity error. In
general, this error is not very large and therefore the extension in the gage
direction (even in multiaxial strain fields) is assumed to be given by
ε =
∆R / R
Sm
Strain gauge measures only extension, that is, the normal component of
strain. It does not measure shear strain directly—this must be obtained by
deduction from a number of extension measurements.
Strain Gage Rosettes
At a particular point on the surface of a deformed structure, there are, in
general, three unknown strains; either ( xx, yy, xy) or 1, 2, ). Therefore,
three independent pieces of experimental data must be recorded if the strain
state is to be specified. Since one strain gauge cannot measure shear directly,
three gauges must be arranged to give three independent extension
measurements and the transformation equations used to determine the shear
component. This combination of three (sometimes four) gauges is called a
strain gage rosette. In practice, rosettes are usually mounted with respect to
a reference axis (not necessarily the principal strain axes).
The majority of rosette gauges are either rectangular rosettes with
and 900 or delta rosettes with = 00, 600 and 1200.
3
= 00, 450
Transformation matrix
The values of the principal strains associated with the three strain readings
taken from a
strain gauge rosette may be found by calculation using the following
transformation eqn.:
This equation can be applied three times for the three values of
of the
rosette gauges. Thus with three known values of for three known values of
, three simultaneous equations will give the unknown strains x, y, xy or
you can use the following transformation matrix:
Or in general:
{
data }
= [ C ]{
coord}.
The principal strains ( and the principal stresses) can then be determined
using the known for you procedure (See Lect.#1).
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