Strain Gage Rosette on Bracket

In order to determine the location and weight of a particular hanging load P, a strain rosette is attached to the top of the bracket near the wall (assume strain gages are at the wall). The rosette has three gages, each are 120^o apart as shown in the diagram. The bracket material is steel, E = 29,000 ksi and ν = 0.29.

When the weight P is placed at a distance s on the rectangular bracket arm, the three gages measure the following strains,

     ε_a = 224.8 μ
     ε_b = -118.3 μ
     ε_c = 132.9 μ

From these three strains, the load P and the distance s need to be determined.

The first step is to transform the strain gage strains to strains that can be represented in a normal x-y strain element. This can be done by rotating the three unknown strains, ε_x, ε_y and γ_xy. into the three known stains, ε_a, ε_b and ε_c. using the basic strain rotation equation,

     ε_x' = (ε_x + ε_y)/2 + (ε_x + ε_y)/2 cos2θ + γ_xy/2 sin2θ

Applying this equation three times, once for each gage where θ_a = 0^o, or θ_b = 120^o, or θ_c = 240^o, gives,

These simplify to

Substituting the first equation into the second two and solving for ε_x and ε_y gives

Substituting the actual gage values gives the final strains in the x-y coordinate system,

     ε_x = 224.8 μ

     ε_y = [2(-118.3 μ) + 2(132.9 μ) - 224.8 μ]/3
         = -65.2 μ

     γ_xy = [-2(-118.3 μ) + 2(132.9 μ)]/1.7321
           = 290.0 μ

Notice, the y-direction normal strain is not zero due to Poisson's effect even though the stress will be zero.

Hooke's law can be used to calculate the x-direction stress and the shear stress.

The load P will cause both a bending moment and a twisting moment at the wall where the strain gages are located. The bending moment is

     M_bending = P (6 in)

This moment will cause a bending stress at the top of the circular bar at the gage location.

     P = 45.0 lb

The twisting moment is

     T_twisting = Ps

This torque will cause a shear stress in the circular bar.

     s = 6.0 in