A spruce timber beam with a rectangular cross$-$section of $10cm20cm$ and a length of $3$ meters is subjected to a concentrated load of $5000$ N $.$ The modulus of elasticity is $($E$)$ for spruce: $11$ GPa $(11{10}^9\frac{N}{m^2}),$ and the allowable deflection is $10$ mm $.$

Using the provided dimensions and information, calculate the deflection of the timber beam when subjected to a concentrated load. Consider the beam to be simply supported at both ends.

Answer the below questions regarding the data analysis.

$($a$)$ Calculate the moment of inertia $($I$)$ for the timber beam.

$($b$)$ Determine the maximum bending stress $()$ induced in the beam.

$($c$)$ Calculate the deflection $()$ of the beam under the applied load.

$($d$)$ Compare the design calculated deflection with the allowable deflection and determine if the beam meets the design criteria.