$1$ a$)$ Explain the following terms briefly:

Limit State Design

Ultimate Limit State

Serviceability Limit State

Permanent Action

Variable Action

$(25$ marks$)$

b$)$ The plan below shows the first floor of a building with a steel column at each corner. The floor consists of a solid, $220mm$ thick, in$-$situ reinforced concrete slab with a $70mm$ thick screed on top.

The floor is divided into two simply$-$supported, one$-$way spanning sections supported by the B$2,$ B$3$ & B$4$ beams only as shown. These three beams span from Gridline $1$ to Gridline $2$ and they all have a self$-$weight of $1k\frac{N}{m}.$

Beams B$2$ & B$4$ are supported directly by the columns whereas beam $B3$ is supported by the two $B1$ type beams

There are two $B1$ type beams which have a self$-$weight of $2k\frac{N}{m},$ span from Gridline A to Gridline D and are supported directly by the columns. These beams directly support their own self$-$weight and the B$3$ beam only.

The variable load on the floor is $3k\frac{N}{m^2}.$ You may assume a density of $25k\frac{N}{m^3}$ for precast concrete and a density of $23k\frac{N}{m^3}$ for the screed.

Calculate the ULS design loading and sketch loading diagrams for beams B$1,$ B$2,$ B$3$ and B$4.$

$(75$ marks$)$