Question $5-$ Pre$-$stressed Concrete Design $(25$ marks$)$

A prestressed concrete girder is simply supported over a span of $20\mathrm{.}00$ m and has the cross section at mid $-$ span as shown in Figure below. During fabrication, the precast beam will be subjected to a superimposed dead load $(G)=6\mathrm{.}5k\frac{N}{m}.$ Assume that the tendon is placed with an eccentricity of $425$ mm at the mid $-$ span section.

Section Properties

$A{g}_g=420{10}^{3}m{m}^2$

$Z_{\text{t}\text{o}\text{p}}=100{10}^{6}m{m}^3$

$Z_{\text{b}\text{o}\text{t}}=110{10}^{6}m{m}^3$

$I_g=65{10}^{9}m{m}^4$

$f^{\text{'}}c=65$MPa

$Q=20k\frac{N}{m}$

$M=1\mathrm{.}1to\frac{n}{m}$

Figure $5\mathrm{.}1$ Mid$-$Span beam Section

a$.$ If the prestressing tendon produces an effective prestressing force $($ F $)$ of $3900$ kN $,(15$ marks$)$ calculate the resulting stresses at the bottom fibre and top fibre of the beam at the mid $-$ span section at transfer and service. Ensure that you draw stress diagrams for your solutions.

b$.$ Show that limiting pre$-$stress requirements have been satisfied for both service and

$(5$ marks$)$ transfer.

C$.$ What is the minimum pre$-$stress forces needed in the top and bottom fiber at midspan

$(5$ marks$)$ to ensure that tensile stress at the bottom fiber is zero.