$($a$)$ An initially straight beam of height $20$ mm is subjected to a moment such that the radius of curvature of the beam is $5$ m $.$ For the plots discussed below: clearly label all curves and axes; graph paper is not necessary; use a ruler or straight edge for the axes or straight lines; the plot does not have to be perfectly to scale.

Note that $Y$ is the coordinate as shown in Figure Q$1$a

$($i$)$ Create a plot of strain vs $Y$

$($ii$)$ For a linearly elastic material with $E=77.$GPa, create a plot of stress vs Y

$($iii$)$ For a linearly elastic, perfectly plastic material with $E=77$GPa and ${}_{\text{y}\text{i}\text{e}\text{l}\text{d}}=0\mathrm{.}1\%,$ create a plot of stress vs $Y$

$[8]$

Figure Q$1$a