E1C11 09/14/2010
13:14:3 Page 483
To remove the effects of bending strain, identical strain gauges are mounted to the top and
bottom of the beam as shown in Figure 11.13, and they are connected to bridge locations 1 and 4
(opposite bridge arms). The gauges experience equal but opposite bending strains (Eq. 11.30), and
both strain gauges are subject to the same axial strain caused by F N . The bridge output under these
conditions is
dE 0
E i
¼
GF
4
e 1 þ e 4
ð
Þ
ð11:31Þ
where e 1 ¼ e a1 þ e b1 and e 4 ¼ e a4 À e b4 , with subscripts a and b referring to axial and bending
strain, respectively. Hence, because e a1 j ¼ e a4 j and e b1 j ¼ e b4 j
j
j
j
j
, the bending strains cancel but the
axial strains sum, giving
dE 0
E i
¼
GF
2
e a
ð11:32Þ
For a single gauge experiencing the maximum strain,
dE 0
E i
¼
GF
4
e a
ð11:33Þ
The ratio of the output represented in Equation 11.32 to that represented in Equation 11.33 has a
value of 2. This is the bridge constant (k ¼ 2) for the strain gauge arrangement in Figure 11.12. The
bending strain cancels from Equation 11.32 indicating that this arrangement compensates for the
bending strain.
A guide for some practical bridge-gauge configurations is provided in Table 11.1.
Temperature Compensation
Differential thermal expansion between the gauge and the specimen on which it is mounted creates
an apparent strain in the strain gauge. So, temperature sensitivity of strain gauges is caused by
temperature changes in the gauge itself and the strain experienced by the gauge as a result of
differential thermal expansion between the gauge and the material on which it is mounted. Using
gauges of identical alloy composition as the specimen minimizes this latter effect. However, even
keeping the specimen at a constant temperature may not be enough to eliminate the effect of gauge
thermal expansion. Heating of the strain gauge as a result of current flow from the measuring device
may be a source of significant error since the gauge is also a temperature-sensitive element. The
temperature sensitivity of a strain gauge is an obstacle to accurate mechanical strain measurement
that must be considered. Fortunately, there are effective ways to deal with it.
Figure 11.14 shows two circuit arrangements that provide temperature compensation for a
strain measurement. The strain gauge mounted on the test specimen experiences changes in
resistance caused by temperature changes and by applied strain, whereas the compensating gauge
experiences resistance changes caused only by temperature changes. As long as the compensating
gauge, as shown in Figure 11.14, experiences an identical thermal environment as the measuring
gauge, temperature effects will be eliminated from the circuit. To show this, consider the case when
all the bridge resistances are initially equal and the bridge is therefore balanced. If the temperature of
the strain gauges now changes, their resistance changes as a result of thermal expansion, creating an
apparent thermal strain. Under an applied axial load, the output of the bridge is derived from
11.6 Apparent Strain and Temperature Compensation 483
13:14:3 Page 483
To remove the effects of bending strain, identical strain gauges are mounted to the top and
bottom of the beam as shown in Figure 11.13, and they are connected to bridge locations 1 and 4
(opposite bridge arms). The gauges experience equal but opposite bending strains (Eq. 11.30), and
both strain gauges are subject to the same axial strain caused by F N . The bridge output under these
conditions is
dE 0
E i
¼
GF
4
e 1 þ e 4
ð
Þ
ð11:31Þ
where e 1 ¼ e a1 þ e b1 and e 4 ¼ e a4 À e b4 , with subscripts a and b referring to axial and bending
strain, respectively. Hence, because e a1 j ¼ e a4 j and e b1 j ¼ e b4 j
j
j
j
j
, the bending strains cancel but the
axial strains sum, giving
dE 0
E i
¼
GF
2
e a
ð11:32Þ
For a single gauge experiencing the maximum strain,
dE 0
E i
¼
GF
4
e a
ð11:33Þ
The ratio of the output represented in Equation 11.32 to that represented in Equation 11.33 has a
value of 2. This is the bridge constant (k ¼ 2) for the strain gauge arrangement in Figure 11.12. The
bending strain cancels from Equation 11.32 indicating that this arrangement compensates for the
bending strain.
A guide for some practical bridge-gauge configurations is provided in Table 11.1.
Temperature Compensation
Differential thermal expansion between the gauge and the specimen on which it is mounted creates
an apparent strain in the strain gauge. So, temperature sensitivity of strain gauges is caused by
temperature changes in the gauge itself and the strain experienced by the gauge as a result of
differential thermal expansion between the gauge and the material on which it is mounted. Using
gauges of identical alloy composition as the specimen minimizes this latter effect. However, even
keeping the specimen at a constant temperature may not be enough to eliminate the effect of gauge
thermal expansion. Heating of the strain gauge as a result of current flow from the measuring device
may be a source of significant error since the gauge is also a temperature-sensitive element. The
temperature sensitivity of a strain gauge is an obstacle to accurate mechanical strain measurement
that must be considered. Fortunately, there are effective ways to deal with it.
Figure 11.14 shows two circuit arrangements that provide temperature compensation for a
strain measurement. The strain gauge mounted on the test specimen experiences changes in
resistance caused by temperature changes and by applied strain, whereas the compensating gauge
experiences resistance changes caused only by temperature changes. As long as the compensating
gauge, as shown in Figure 11.14, experiences an identical thermal environment as the measuring
gauge, temperature effects will be eliminated from the circuit. To show this, consider the case when
all the bridge resistances are initially equal and the bridge is therefore balanced. If the temperature of
the strain gauges now changes, their resistance changes as a result of thermal expansion, creating an
apparent thermal strain. Under an applied axial load, the output of the bridge is derived from
11.6 Apparent Strain and Temperature Compensation 483