Defining and measuring strain: part 2

In part 1 of this series, we looked at the strain gauge, which can be used to quantify how a test specimen deforms as a function of applied stress. We developed a Wheatstone-bridge circuit that makes use of strain-gauge elements in two of the bridge’s legs, as shown in Figure 1, a version of Figure 3 from part 1. Note that the strain-gauge elements take the place of resistors R₂ and R_X in Figure 2 from part 4 of our previous series on Kirchhoff’s laws.

Q: In Figure 1, it looks like you’ve changed some of the labels.
A: Right. I had been using V_DMM to emphasize that we are using a modern digital multimeter (DMM) instead of a 19th-century galvanometer, which would have been used in early Wheatstone bridges. 19th-century galvanometers were quite useful in detecting zero current and voltage conditions, with applications extending to the decoding of transmitted signals in early trans-Atlantic telegraphy. However, such galvanometers were not good at quantifying non-zero levels, which modern instruments can do easily and accurately. In Figure 1, I’m using labels commonly found in strain-gauge literature. For example, I’ve changed V_DMM to V_O, for output voltage. In addition, I changed V_IN to V_EX, for excitation voltage. Finally, in the setup from our earlier series, V_DMM varied inversely with R_X. In Figure 1, I have reversed the meter polarity, so V_O varies directly with R_X and hence strain e.

Figure 1. This figure includes labels commonly found in strain-gauge literature, including VEX for excitation voltage and VO for output voltage. (Image: Rick Nelson)

To review, in Figure 1, R_X is an active strain gauge, and R₂ is a dummy strain gauge used for temperature compensation. Note that the long, thin wires of R₂ are mounted perpendicular to the direction of tension, so their resistance is relatively unaffected by the strain.

Q: So, how do we calculate strain given V_O?
A: From our previous series, we demonstrated how to calculate R_X based on the voltage reading, with the DMM or output voltage equaling V_V – V_X. For the configuration shown in Figure 3 with the polarity switch,

We can rearrange this equation as follows:

Then we can calculate R_X as a function of V_O:

Q: And how does strain relate to R_X?
A: From part 1 of this series, we know that given a gauge factor GF, strain is

We also see that DR is R_X – R₂, where R₂ equals the unstrained resistance of R_X, or 120 W. So, given V_O and a GF value of 2, we can directly calculate strain:

Figure 2 plots this equation for values of V_O from -10 mV to +10 mV.

Figure 2. Given the VO for the Figure 1 circuit, we can derive the strain. (Image: Rick Nelson)

Figure 3. Two active strain gauges on a test specimen are aligned with the applied stress. (Image: Rick Nelson)

Q: Can we design a bridge configuration where both strain-gauge elements are active, and if so, what would be the advantages?
A: Figure 3 shows an alternative arrangement to Figure 1, with one strain gauge depicted in red placed at the top of a test specimen, while another depicted in blue is placed at the bottom, with the sense wires in both aligned with the direction of stress, so both are active.

Next time, we’ll take a look at how to apply this configuration and describe its benefits.

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