Two Dimensional Strain System

Q. Given that for an element in a body of homogeneous isotropic material subjected to plane stresses, Ex, ${\epsilon }_x,{\epsilon }_y$ and ${\epsilon }_z$ are normal strains in x, y and z direction respectively and μ is the Poisson's ratio, the magnitude of unit volume change of the element is given by

1. ${\epsilon }_x+{\epsilon }_y+{\epsilon }_z$
2. ${\epsilon }_x+\mu ({\epsilon }_y+{\epsilon }_z)$
3. $\mu ({\epsilon }_x+{\epsilon }_y+{\epsilon }_z)$
4. $\frac{1}{{\epsilon }_x}+\frac{1}{{\epsilon }_y}+\frac{1}{{\epsilon }_z}$

Q. In a plane strain situation in xy plane, the displacements at a point are given as:

$u=(-2x+8y)\times {10}^{-6}unit$

$v=(-3x+5y)\times {10}^{-6}unit$

What is the shearing strain?

1. $9\times {10}^{-6}$

2. $7\times {10}^{-6}$
3. $5\times {10}^{-6}$
4. $3\times {10}^{-6}$

Q. Assertion (A): A plane stress system consists of two principal stresses ${\sigma }_{1}and{\sigma }_2$ and a plane strain system consists of two principal strains ${\epsilon }_1={\sigma }_1/Eand{\epsilon }_2={\sigma }_2/E$. Both the systems are identical.
Reason (R): Stress is proportional to strain.

1. both A and R are true and R is the correct explanation of A
2. both A and R are true but R is not a correct explanation of A
3. A is true but R is false
4. A is false but R is true

Q. In a biaxial strain system ${\epsilon }_{x}and{\epsilon }_y,$ what is the maximum engineering shearing strain?

1. ${\epsilon }_x+{\epsilon }_y$

2. ${\epsilon }_x-{\epsilon }_y$

3. $\frac{{\epsilon }_x+{\epsilon }_y}{2}$

4. $\frac{{\epsilon }_x-{\epsilon }_y}{2}$

Q. An element of a certain material in plane strain has
${\epsilon }_x=800\times {10}^{-6}$

${\epsilon }_y=400\times {10}^{-6}$

${\gamma }_{xy}=300\times {10}^{-6}$
What is the maximum shearing strain?

1. $150\times {10}^{-6}$
2. $355\times {10}^{-6}$
3. $250\times {10}^{-6}$
4. $500\times {10}^{-6}$