I was having trouble finding the "minimum required cross$-$sectional properties of Beam AB to satisfy both the ultimate $($i$.$e$.$ strength$)$ and serviceability $($i$.$e$.$ deflection limit states for the beam" which then made it hard to complete the second part of part B$.$ Can you please show both the moment$-$area theorems and the virtual work principle for part B$.$ THANKS

Question $3(33$ marks$)$

A steel framing plan that supports a reinforced concrete floor is shown in Figure $2.$ The reinforced

concrete floor is $200mm$ thick; the self weight of the steel beams is equivalent to a uniformly

distributed surface load of $0\mathrm{.}3k\frac{N}{m^2}$; the superimposed dead load on the floor is $0\mathrm{.}5k\frac{N}{m^2},$ and

the specified live load on the floor is $2\mathrm{.}4k\frac{N}{m^2}.$ Assume the unit weight of reinforced concrete to

be $24k\frac{N}{m^3}.$

Figure $2$

$[15$ marks$]$ a$)$ Develop the shear force and bending moment diagrams for Beam AB under the total

factored dead and live loads based on a dead load factor of $1\mathrm{.}25$ and a live load factor of $1\mathrm{.}5.$

$[18$ marks$]$ b$)$ Determine the minimum required cross$-$sectional properties of Beam $AB$ to satisfy

both the ultimate $($i$.$e$.$ strength$)$ and serviceability $($i$.$e$.$ deflection$)$ limit states for the beam. The

deflection control criterion requires that the maximum deflection of Beam $AB$ under the specified

live load not exceed $40mm.$ Assume ${}_y=350$MPa,${}_y=350$MPa,$E=2\mathrm{.}0{10}^5$MPa, resistance

factor $=0\mathrm{.}9,$ and cross$-$sectional properties to be constant throughout the beam.

Note: Either the moment$-$area theorems or virtual work principle may be used to compute the

maximum deflection of Beam $AB.$ In either case, the appropriate bending moment diagram$($s$)$

MUST be drawn.