Problem $11.$ The $60$ strain rosette is mounted on the surface of the bracket. The following readings are obtained for each gage: ${}_a=-780({10}^{-6}),{}_b=400({10}^{-6}),$ and ${}_c=500({10}^{-6}).$

Determine $($a$)$ the principal strains and $($b$)$ the maximum in$-$plane shear strain and associated average normal strain. In each case show the deformed element due to these strains.

$**$

Note: The strain components could be determined using the below strain transformation equations.

${}_a={}_{x}co{s}^2{}_a+{}_{y}si{n}^2{}_a+{}_{xy}sin{}_{a}cos{}_a$

${}_b={}_{x}co{s}^2{}_b+{}_{y}si{n}^2{}_b+{}_{xy}sin{}_{b}cos{}_b$

${}_c={}_{x}co{s}^2{}_c+{}_{y}si{n}^2{}_c+{}_{xy}sin{}_{c}cos{}_c$