Double integration of acceleration over time
Double integration with an accelerometer would not yield any useful results under static loading.
What you could do instead is the following (the longer the beam the better - a bridge would be ideal for this):
place a few accelerometers along one side of the beam (if you can place on top and bottom even better). You'd also need to know the exact position along the beam for each accelerometer.
from each accelerometer, you estimate the change in angle. you need to take two measurements :
- no load
- static load (apply the load, wait to stabilise and then start to measure).
Since load is static you can measure over a long time and get a quite accurate measurement. Essentially what you are measuring is the rotation of the gravity vector.
Provided the load is uniform, or concentrated, it is now possible to perform a high order least squares fit on the angle w.r.t. to position along the beam $w'(x)$. ($w(x)$: the lateral displacement).
Then it would be possible to interpolate over the position with - I expect - meaningful results.