In a light industrial building, a simply supported rectangular Precast Prestressed Concrete Beam $($PPCB$)$ of $600mm$ width $x400mm$ depth and spanning $8m,$ is supporting Precast Pretensioned Hollow$-$Core Slab $($HCS$),200mm$ thick with span length $6m.$ Assume all the concrete grade used is C$40/50$ to Eurocode $2$ design. The support condition is unpropped. The stages of construction for the PPCB at critical midspan are shown below in Figure Q$-2$:

$($a$)$Stage $1$: Construction Stage

Placement of HCS on the rectangular PPCB and casting of concrete infill into the HCS$.$

Unfactored self$-$weight of HCS

$($Including cast in$-$situ infill

& construction live loads$)=4\mathrm{.}0$kPa

Unfactored self$-$weight of the PPCB

$($b$)$Stage $2$: Final$/$Service$/$Operation

Stage

Placement of superimposed dead load $($SDL$)$ due to partition, finishes, M&E services on $HCS=2\mathrm{.}5$kPa

Live Load $=3\mathrm{.}0$kPa

Figure Q$-2$ Midspan Section View of $($a$)$ Construction Stage $1$ and $($b$)$ Final Stage $2$ of a composite PPCB with $1\mathrm{.}2m$ effective width of flange built from HCS

The sectional properties at midspan are shown below:

Moment of inertia of the rectangular PPCB$,I_{co}=3200{10}^{6}m{m}^4$

Distance from centroid of rectangular PPCB to soffit$/$ bottommost fiber of the beam, $y_b=190mm.$

Moment of inertia of the composite PPCB$,I_{cc,co}=8457{10}^{6}m{m}^4$

Distance from centroid of the composite PPBC to soffit $/$ bottommost fiber of the beam, $y_{bc,co}=378mm.$

It is given that the composite PPCB has a final stress after accounting for prestress losses at its bottom fibre $({}_b)$ of $13\frac{N}{m}{m}^2($compression$)$ and topmost fibre $({}_t)$ of $-4\frac{N}{m}{m}^2($tension$).$

$($a$)$ Show that the Stage $1$ and Stage $2$ service moment due to loadings acting at midspan of the PPCB$,$ are for Stage $1)$ and for Stage $2)$

$[4$ Marks$]$

$($b$)$Compute the following section properties for the PPCB and its composites

$($i$)$ Section modulus of rectangular PPCB at top fibre, $Z_{t,co}$

$($ii$)$ Section modulus of rectangular PPCB at bottom fibre, $Z_{b,co}$

$($iii$)$ Section modulus of the composite PPCB section at bottom fibre, $Z_{bc,co}$

$($iv$)$ Section modulus of the composite PPCB section at top of the rectangular PPCB$,Z_{t,co}$

$($v$)$ Section modulus of the composite PPCB section at topmost fibre, $Z_{\text{'}\text{t}\text{e}\text{c}\text{o}}$

$[10$ Marks$]$

$($c$)$ Check and comment the PPCB composite section for allowable permissible service stresses for combined Stage $1$ and Stage $2$ loadings based on the following EC$2$ criteria. From EC$2$ Table $3\mathrm{.}1,$ mean concrete tensile strength, $f_{ctm}=3\mathrm{.}5$MPa.

$($i$)$ magnitude of bottom fibre stress of PPCB composite,

$[4$ Marks$]$

$(ii)$ magnitude $of$ top fibre stress $of$ composite $at$ top $of$ rectangular $PPCB,f_t<mn>0</mn><mn>.</mn><mn>4</mn><mn>5</mn><msub><mrow><mi>f</mi></mrow><mrow><mi>c</mi><mi>k</mi></mrow></msub><mtext></mtext><mi>[</mi><mn>4</mn>$ Marks$]$

$(iii)$ magnitude $of$ top fibre stress $ofat$ topmost fibre $of$ composite $PPCB,f^{\text{'}}<mtext></mtext><mn>0</mn><mn>.</mn><mn>5</mn><mn>6</mn><mn>7</mn><msub><mrow><mi>f</mi></mrow><mrow><mtext>c</mtext><mtext>k</mtext><mtext>i</mtext></mrow></msub>$