Problem $2\mathrm{.}3-4$ A rectangular bar of length $L$ has a slot in the

middle half of its length $($see figure$).$ The bar has width $b,$

thickness $t,$ and modulus of elasticity $E.$ The slot has width $\frac{b}{4}.$

$($a$)$ Obtain a formula for the elongation $$ of the bar due

to the axial loads $P.$

$($b$)$ Calculate the elongation of the bar if the material is

high$-$strength steel, the axial stress in the middle

region is $160$ MPa $,$ the length is $750$ mm $,$ and the

modulus of elasticity is $210$ GPa $.$

$($c$)$ If the total elongation of the bar is limited to

${}_{\text{m}\text{a}\text{x}}=0\mathrm{.}475mm,$ what is the maximum length of the slotted region? Assume that the axial stress in the middle region

remains at $160$ MPa $.$