$1$ A simply supported beam whose density is $800k\frac{g}{m^3}$ has a total span of $L=12$ meters,

a$.$ Determine the dimensions of a rectangular beam $(b2b)$ which can be used without exceeding a bending stress of $25$ MPa due to its own weight only. Round up your answer to the nearest $25-$mm increment.

b$.$ Using the beam dimensions in $($a$),$ how much additional uniformly distributed load $($UDL$)$ can it carry if the allowable bending stress is increased to $30$MPa $?$

c$.$ If we use a built$-$up rectangular cross$-$section made up of five $10025mm$ wood planks nailed together as shown in Figure $1.$ What must be the maximum spacing of the nails to carry both its own weight and the additional weight from $($b$)?$ The strength of each nail is $2\mathrm{.}5N.$

A$.$ If Figure $1$ is re$-$arranged to form the cross section shown in Figure $2,$ what will now be the maximum moment and maximum shear that it can carry?

e$.$ What is the shear stress developed in each nail of Figure $2?$

$-$f$.$ If Figure $1$ is re$-$arranged to form the cross section shown in Figure $3,$ what will now be the maximum moment and maximum shear that it can carry?

$.$g$.$ What is the shear stress developed in each nail of Figure $3?$

h$.$ If the rectangular cross section in $($a$)$ is replaced by a triangular one, whose base $=b$ and height $=2b,$ what is the maximum shearing stress that is developed in this triangular beam?

Figure $1$

Figure $2$

Figure $3$