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In Section 4.2.1, we saw that the resistance of a metal sample is given by
where Ro = sample resistance W
p = sample resistivity W-m
lo = length in m
Ao = cross-sectional area in m2
Suppose this sample is now stressed by the application of a force F as shown in Figure 5.12a. Then we know that the material elongates by some amount Dl so that the new length is l = l + Dl. It is also true that in such a stress-strain condition, although the sample lengthens, its volume will remain nearly constant. Because the volume unstressed is V = loAo, it follows that if the volume remains constant and the length increases, then the area must decrease by some amount DA:
Because both length and area have changed, we find that the resistance of the sample will have also changed:
Using Equations (5.9) and (5.10), the reader can verify that the new resistance is approximately given by
from which we conclude that the change in resistance is
Equation (5.12) is the basic equation that underlies the use of metal strain gauges because it shows that the strain Dl/l converts directly into a resistance change.
EXAMPLE 5.6
Find the change in a nominal wire resistance of 120 W that results from a strain of 1000 mm/m.
Solution
We can find the change in gauge resistance from
Example 5.6 shows a significant factor regarding strain gauges. The change in resistance is very small for typical strain values. For this reason, resistance change measurement methods used with strain gauges must be highly sophisticated.
Measurement Principles
The basic technique of strain gauge (SG) measurement involves attaching (gluing) a metal wire or foil to the element whose strain is to be measured. As stress is applied and the element deforms, the SG material experiences the same deformation, if it is securely attached. Because strain is a fractional change in length, the change in SG resistance reflects the strain of both the gauge and the element to which it is secured.
Temperature Effects
If not for temperature compensation effects, the aforementioned method of SG measurement would be useless. To see this, we need only note that the metals used in SG construction have linear temperature coefficients of µ@ 0.004/°C, typical for most metals. Temperature changes of 1°C are not uncommon in measurement conditions in the industrial environment. If the temperature change in Example 5.6 had been 1°C, substantial change in resistance would have resulted. Thus, from Chapter 4,
or
DRT = RoµDT
where DRT = resistance change because of temperature change
µo @ 0.004/°C in this case
DT @ 1°C in this case
R(To) = 120 W nominal resistance
Then, we find DRT = 0.48 W, which is twice the change because of strain! Obviously, temperature effects can mask the strain effects we are trying to measure. Fortunately, we are able to compensate for temperature and other effects, as shown in the signal conditioning methods in the next section.
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Related Links:
Strain and Stress
Metal Strain Gauges
Semiconductor Strain Gauges
Load Cells
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