Problem $6$

A strain gauge measures the normal strain along its length. The strain gauges $a,b,$ and $c$ pictured below

measure strains ${}_a=-4{10}^{-3},{}_b=2{10}^{-3},$ and ${}_c=3{10}^{-3}$ in orientations of $0,45,$ and $90$ relative

to the $x$ axis.

Determine

a$.$ the shear strain component ${}_{xy},$

b$.$ the normal strain oriented $60$ relative to the $x$ axis $($with positive defined as counterclockwise$),$

c$.$ the shear strain between two orthogonal directions at $60$ and $-30$ relative to the $x$ axis,

d$.$ the in$-$plane principal strains and their directions, and

e$.$ the maximum shear strain.