$00124$

Problem $1.(7$P$)$

The piece of rubber is originally rectangular. Determine the average shear at A if the corners $B$ and $D$ are subjected to the displacements that cause the rubber to distort as shown by the dashed lines. Determine the average normal strain along the diagonal OB knowing that deformed position of points D and B are $D^{\text{'}}$ and $B^{\text{'}},$ respectively.

Step I Using small deformation concept find angle in radians of vertical edge from A with respect toy

Step $2$ Find angle in radians of horizontal edge from A with respect to $x$

Step $3$ Shear strain at $A$ is sum of angles from Step $1$ and Step $2$

Step $4$ Find angle B'AD'

Step $5$ Find lengths of $A{D}^{\text{'}},A{B}^{\text{'}}$ and $DB$

Step G find D$\text{'}$B$\text{'}$ using law of cosines

Step $7$ Find Normal strain along $0B$ using the formula: strain $=\frac{0^{\text{'}}{B}^{\text{'}}-DB}{D}B$