Reddit reviews Theory and Design for Mechanical Measurements

As you probably already know, strain gauges change their resistance as they are strained / deformed. They are merely a metallic wire wrapped back and forth. When bent inward or compressed, the wire thickens and the resistance decreases. When bent outward or in tension, the wire thins and resistance increases.

http://i.imgur.com/RNEfY22.png
Image Source

That's fantastic, except, resistance is not very nice to measure over time. Sure you can measure it with a multimeter, but plunking that into a computer that can capture data over time isn't as clean.

Voltage, however, is very easy to measure over time. To do that, they use something called a wheatstone bridge. A simple diamond of resistors that allows one to detect a small chain in resistance as a voltage.
http://i.imgur.com/i7R0OFp.png Image Source

Where in place of some of the resistors, you instead place your strain gauge. A variable resistor.

Gauges can be mounted in any number of configurations. Here are a few common ones.
http://i.imgur.com/HzbuTuF.png Image Source

Depending on how they are mounted they allow one to compensate for other variability that may be recorded by the strain gauge that isn't what you wish to measure.

Great! Now let's take configuration 1 and do some maths.

A single strain gauge, on a beam in uniaxial stress.

You can derive the wheatstone bridge equations if you wish, but what it comes down to is:
http://i.imgur.com/YhtKo8w.png Image Source

Let's put one strain gauge on leg X. (Effectively leg 4) Replace X with R + delta R because it is in tension. Let's also assume the resistance of the other legs of the bridge is the same as the nominal resistance of the gauge. (A fair assumption, as you'd likely do that if you made your own bridge.)
http://i.imgur.com/6iwkETC.png Image Source

Delta R is a function of strain.
http://i.imgur.com/lFQqKIJ.png Image Source

For these types of strain gauges, GF, gauge factor, is 2. (approximately)

Let's solve that equation for delta R and plug it back into our voltage eqn.
http://i.imgur.com/NpTt8ki.png Image Source

Really simplifies down. Cool.

Except you rarely get off that easily; you also rarely use only 1 strain gauge. Let's try a half bridge, 2 strain gauges, then do a full bridge, all 4 legs are gauges.

Back to that table, we can try the second one. Two strain gauges in bending, one measuring compression, the other tension. Placed on the left side legs of the bridge.

Back to the wheatstone formula, except this time leg 1 is R + delta R because it is in tension and leg 3 is R - delta R because it is in compression.
http://i.imgur.com/Rarqwel.png Image Source

Lastly, let's step up to the full bridge. Instead of configuration 4, where it is just a beam in tension, let's bend it instead. Where all legs are strain gauges, let's throw 1 and 4 on the bottom so they are in compression R = R - dR, and 2 and 3 on top so they are in tension, R = R + dR.

http://i.imgur.com/SDLwxdb.png Image Source

If we did something like configuration 4 in the table, we'd record close to nothing in bending. As it states in the compensation provided - effects from bending are eliminated from the calculations.

Let me know if you'd like to see more examples. Also double check my calculations, I just did those up quickly in mathcad to show.