$2.[$Horizontal shear stress in beam$](36$ pt$)$

A beam AC is simply supported and carries a

uniformly distributed load within AB segment

$($loading density $20$ kN$/$m$).$ The loading is shown in

schematic $($a$).$

$(1)$ Please find the maximum vertical shear stress and

the corresponding location in the beam. $(5$ pt$)$

$(2)$ We first assume that the beam is made of single

material $($wood$)$ with Young$$s modulus of $12\mathrm{.}5$GPa.

The cross$-$section area of the beam is shown in

schematic $($b$).$

For the cross$-$section area shown in $($b$),$ please

determine the second moment of area with respect to

its neutral axis. $(5$ pt$)$

There are two interfaces D and E shown in schematic

$($c$).$ Please determine the horizontal shear stresses on

interfaces D and E in the beam at the position with the

maximum vertical shear stress. $(10$ pt$)$

$(3)$ Then, we assume that the beam is made of two

materials: wood $($Young$$s modulus $12\mathrm{.}5$ GPa$)$ and

steel $(100$ GPa$).$ The cross$-$section area is shown in

schematic $($d$).$ We would like to determine the

horizontal shear stresses on interfaces D and E at the

position B$.$

Please use the following steps:

Step $1$: Please transform steel to wood to obtain an allwood cross$-$section area. Please determine the second

moment area of the all$-$wood cross$-$section area. $(4$ pt$)$

Step $2$: On the all$-$wood cross$-$section area, please

calculate the horizontal shear stresses at interfaces D

and E$.$ Please use the vertical shear stress at position

B$.(8$ pt$)$

Step $3$: Transform the cross$-$section area back to the

composite cross$-$section to obtain the horizontal shear

stresses on the interfaces D and E at position B$.(4$ pt$)$