Q$1$ A straight$-$wire pre$-$tensioned $200$mm x $400$mm beam in C$50/60$ concrete spans $7$m

simply$-$supported. In addition to its own self$-$weight, it carries a super$-$imposed dead

load of $5$kN$/$m and an imposed live load of $9$kN$/$m$,$ all loading being at SLS and

uniformly$-$distributed.

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a$)$ Calculate the central deflections at transfer and in$-$service for the beam described in

Q$1$ above. Take Et $=37$kN$/$mm$2,$ and Es $=15$kN$/$mm$2.$

b$)$ Calculate and illustrate the incremental build$-$up of stress $($separately due to S$/$W$,$

DL and LL$)$ for both transfer and in$-$service stages and at both support and mid$-$span

positions. Take P $=750$kN$,$ e $=+70$mm $($down from centre$)$ and k $=0\mathrm{.}75.$

c$)$ Do the stresses exceed the recommended limiting concrete stresses $($material

strengths$)$ defined in EC$2,$ assuming a transfer strength of $2/3$ of the $28-$day

strength?

d$)$ Set up an Excel spreadsheet to calculate the support and mid$-$span top and bottom

fibre stresses for both design stages using all the key variables $($section size and

self$-$weight; applied DL and LL; span; pre$-$stress$).$ Using your Q$1$ hand calculations

to validate your spreadsheet. Undertake a thorough parametric study of how

changing a single variable at a time on the Q$1$ base$-$case effects the stresses.

Summarise your findings.