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I got a question thats been bugging me for a while:

Say that you're given a straight square beam (say one meter long), and you attach three strain gauges on each face. One at 0 degrees, one at 45 degrees and one at 90 degrees. All of these gauges are flat against the beam, so they measure in plane.

My question is: given these strains, could you reconstruct the geometry?

Assumptions:

  • The material is linear
  • The deformations are small

I've been trying it for a while, and I can always resolve for the compression and curvature, but the rotation seems to elude me. For simple cases (like pure bending) it works fine, the sensor at 45 degrees are linearly correlated to the bending. However, when shear is present, these seems not to be the case anymore. As of yet, I can't find a way to correct for this. Anyone got an idea? This problem has probably already been solved by someone else, but I can't seem to find a good reference to this (for the 3D case).

Any help is greatly appreciated!

Edit: images included

Front view of a small section of beam. The red images indicate strain gauges. As can be seen, only strains in plane can be measured.

Front view of a small section of beam. The red lines indicate strain gauges. As you can see, only the strains in plane can be measured

Isometric view of a small section of beam, showing two strain gauges as well as the axis. The torsion mentioned would work on the y axis.

Isometric view of a small section of beam, showing two strain gauges as well as the axis.

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  • $\begingroup$ I think the OP is trying to say that the strain gauges themselves are rotated with respect to the axis of the beam - in other words, one is oriented parallel to the axis, one is 45-deg to the axis, and the other perpendicular to the axis (pictorially, -- / | ) $\endgroup$
    – grfrazee
    Commented Oct 14, 2015 at 14:34
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    $\begingroup$ What is meant by "geometry"? The cross-section of the beam? Its span? Its boundary conditions? $\endgroup$
    – Wasabi
    Commented Oct 14, 2015 at 16:57
  • $\begingroup$ As I see you want to get the beam section geometry corresponding to the strain and torque. It may be solved (in CAE for example) if you know section form and some dimensions (i.e. you know square side and want to find the thikness). In general case, I think that there are more then one solution (large side - small thikness and vice versa) $\endgroup$
    – user3361
    Commented Oct 14, 2015 at 18:22

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TL;DR You don't have enough measurements

Took a while, but it can be proven that (using only the strains on the sides), that there is no way to calculate the resulting shape. The reason for this is the (nonlinear) influence of shear and the lack of constraints.

The other way around (from displacement to strain) is possible, which is why programs like ANSYS can find a result.

Luckily, in most cases, the shear is negligible, in which case you can describe the shape as a combination of Torsion, Compression and Curvature (both x and y) from the given measurements.

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