Current Attempt in Progress

A wooden I$-$beam is fabricated from three pieces of dimension lumber. The cross$-$sectional dimensions are $b_f=86mm,t_f=42mm,d_w=231mm,$ and $t_w=42mm.$ The $1-$beam will be used as a simply supported beam to carry a concentrated load $P$ at the center of a $3m$ span. The wood has an allowable bending stress of $8100$kPa and an allowable shear stress of $680$kPa. The flanges of the beam are fastened to the web with screws that can safely transmit a force of $750N$ in direct shear.

$($a$)$ If the screws are uniformly spaced at an interval of $s=190mm$ along the span, what is the maximum concentrated load $P$ that can be supported by the beam based on the strength of the screw connections? Demonstrate that the maximum bending and shear stresses produced by $P$ are acceptable,

$($b$)$ Determine the magnitude of load $P$ that produces the allowable bending stress in the span $($i$.$e$.,{}_b=8100$kPa $).$ What screw spacing $s$ is required to support this load magnitude?