Question

The components of strain tensor at a point in the plane strain case can be obtained by measuring longitudinal strain in following directions?

• A. along any two arbitrary directions
• B. along any three arbitrary directions
• C. along two mutually orthogonal directions
• D. along any arbitrary direction

Answer 1

The correct option is B along any three arbitrary directions

In plane strain case, component of strain tensor at a point are ${ϵ}_x,{ϵ}_y,{\gamma }_{xy}$
We have threee unknown so we require 3 equations to find them. These unknowns are related to longitudinal strain by the equation.
${ϵ}_1={ϵ}_x{\cos }^2{\theta }_1+{ϵ}_y{\sin }^2{\theta }_1+\frac{{\gamma }_{xy}}{2}\sin 2{\theta }_1$
${ϵ}_2={ϵ}_x{\cos }^2{\theta }_2+{ϵ}_y{\sin }^2{\theta }_2+\frac{{\gamma }_{xy}}{2}\sin 2{\theta }_2$
${ϵ}_3={ϵ}_x{\cos }^2{\theta }_3+{ϵ}_y{\sin }^2{\theta }_3+\frac{{\gamma }_{xy}}{2}\sin 2{\theta }_3$
Hence to find out component of strain tensor we measure longitudinal strain $({ϵ}_1,{ϵ}_2,{ϵ}_3)$ along any three arbitrary direction.