I've become interested recently though an honours project by the construction of six degree of freedom load cells using strain gauges. In many cases I believe this information is unpublished since the load cell manufacturers have no desire or need to publish such information. There are some published on various designs, however, and they vary considerably in terms of mechanical design and strain gauge arrangement.

I've heard that high-end load cells use up to 32 strain gauges to resolve the six forces and torques. In such arrangements, four or more strain gauges are placed in groups carefully designed to reject off-axis loads. Six directions and four per group make 24, which is the number used by Joo et al. (2002), whose device consists of a steel rectangular shaft with multiple cut-outs to provide mobility for the various strain gauge groups.

A simpler mechanical system that also used 24 strain gauges was developed here at The University of Adelaide, described by Dr Carl Howard in his thesis ‘Active isolation of machinery vibration from flexible structures’.

Still, using 24 strain gauges to measure just 6 output signals could be considered overkill. Bayo and Stubbe (1989) investigated a truss arrangement on the assumption that truss elements can only undergo one main mechanism of deformation, axial strain, and therefore would be much easier to instrument using strain gauges. Their design uses only eight but the truss does not look easy to machine — except now with 3D printing this may no longer be an issue.

Finally, the more mathematically-minded rather than mechanical-minded may question why more than six strain gauges are needed anyway. Surely with a particular geometry and understanding of the strain kinematics, six load signals could be calculated from six strain measurements. Indeed, this idea was investigated by Dai and Kerr (2000) using a Steward platform (also known as a hexapod), in which the ‘legs’ of the platform were tension cables mounted to cantilevered strain gauge beams.

I'm not aware of any work that compares these various approaches for different requirements. The truss-like structures seem promising, but are they structurally rigid enough for dynamic loading situations? It's intuitive that the devices with fewer strain gauges could suffer from more cross-axis contamination, but if the system is well characterised perhaps that can be well accounted for. The savings in design complexity, cost of strain gauges and the cost associated electronics certainly deserve the comparison.

Bayo and Stubbe (1989) "Six-axis force sensor evaluation and a new type of optimal frame truss design for robotic applications." Journal of Robotic Systems 6. DOI: 10.1002/rob.4620060206

Dai and Kerr (2000) "A six-component contact force measurement device based on the Stewart platform." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering 214. DOI: 10.1243/0954406001523696

J.W Joo, K.S Na, D.I Kang (2002) "Design and evaluation of a six-component load cell." Measurement 32. DOI: 10.1016/S0263-2241(02)00002-7