Consider a beam with depth $($W$)$ and span $($S$).$ The beam is subjected to a point load $($P$)$ and has a mid$-$span crack with depth and width of $($a$).$ The beam width is B$.$ Verify that the fracture toughness $($K$)$ can be expressed as a function of crack length $($a$)$ for plane stress condition as follows K$=$P$/2$B $(3/2[(-($S$-$a$)^2)/$W$^3+($S$^3-($S$-$a$)^3)/($W$-$a$)^4+($S$-$a$)^2/($W$-$a$)^3)$

Hint: Use the following expression to describe the bending stiffness EI

EI$=\{(($EBW$^3)/12$ for xS$-$a$)/2$@$($EB$($W$-$a$)^3)/12$ for $($S$-$axS$/2)$

Where x is the distance along the beam. Neglect all shear deformations.