For the delta strain gauge rosette alpha1 0 alpha2 60degre

For the delta strain gauge rosette (alpha_1 = 0, alpha_2 = 60degree, alpha_3 = 120degree), show that epsilon_xy =1/square squareroot3?(epsilon60degree-epsilon120degree).

Solution

the known equation is:
x\' = xcos2 +ysin2 + 2 xysin cos

we have, 2 = 60o, so the above equation becomes as follows:
       60= xcos260 +ysin260 + 2 xy sin60cos60

        60= 1/4*x +3/4*y + 3/2*xy

we have, 2 = 120o, so the first equation becomes as,

     120= xcos2120 +ysin2120 + 2 xy sin120cos120

      120= 1/4*x +3/4*y - 3/2*xy

solve the equations 60= 1/4*x +3/4*y + 3/2*xy and 120= 1/4*x +3/4*y - 3/2*xy

we get the following relation:
      xy = 1/3* (₆₀ -₁₂₀)