Q$.1$ A $24\mathrm{.}0m$ span, simply$-$supported post$-$tensioned prestressed concrete footbridge is to be designed, as shown in Figure Q$\mathrm{.}1.$

Using the data provided:

$($a$)$ check whether the cross$-$section of the beam is appropriate by verifying through calculations that $Z_i$ and $Z_b$ are larger than the required;

$($b$)$ determine the permissible tendon zone through calculations at mid$-$span, quarter$-$spans and supports, and plot the calculated results on the graph paper provided, where only the tendon zone for half the span is required due to symmetry.

$(30)$

Design data

Total marks $[40]$

Unit weight of concrete $(r$ ano$)$

Self$-$weight allowance for the deck $($excluding the beam$)=25k\frac{N}{m^3}$

Superimposed dead loading due to surfacing $=9k\frac{N}{m^2}$

Imposed loading $=3k\frac{N}{m^2}$

Maximum allowable concrete stress at transfer $(f_{\text{m}\text{a}\text{x}}^{\text{'}}),=,7k\frac{N}{m^2}$

Minimum allowable concrete stress at transfer $(f_{\text{m}\text{i}\text{n}}),=18\mathrm{.}8\frac{N}{m}{m}^2$

Maximum allowable concrete stress at service $(f_{\text{m}\text{a}\text{x}}),=-1\mathrm{.}0\frac{N}{m}{m}^2$

Minimum allowable concrete stress at service $(f_{\text{m}\text{i}\text{n}}),=24\mathrm{.}0\frac{N}{m}{m}^2$

Initial prestress force in the tendon $(P_i),=0\mathrm{.}0\frac{N}{m}{m}^2$

Short$-$term prestress loss $(1-),=11000kN$

Long$-$term prestress loss $(1-),=,12\%$

$,=,26\%$

Note: Superimposed dead loading due to surfacing should be considered at service stage.

$Z_i\frac{M_i-M_i}{f_{\text{m}\text{a}\text{x}}-f_{\text{m}\text{i}\text{n}}}$

$Z_b\frac{M_3-M_i}{f_{\text{m}\text{a}\text{x}}-f_{\text{m}\text{i}\text{n}}}$

$e\frac{Z_t}{A_t}+\frac{1}{P_i}(M_i-Z_t{f}_{\text{m}\text{i}\text{n}}^{\text{'}})$

$e\frac{1}{P_i}(M_i+Z_b{f}_{\text{m}\text{a}\text{x}}^{\text{'}})-\frac{Z_b}{A_t}$

$e\frac{Z_t}{A_t}+\frac{1}{P_1}(M_s-Z_t{f}_{\text{m}\text{a}\text{t}})$

$e\frac{1}{P_1}(M_s+Z_b{f}_{ma})-\frac{Z_b}{A_t}$

$e\frac{1}{P}(M_i+Z_0{f}_{\text{m}\text{a}\text{x}})-\frac{Z_b}{A_t}$

$e\frac{Z_t}{A_t}+\frac{1}{P_1}(M_s-Z_1{f}_{\text{m}\text{a}\text{x}})$

$e\frac{1}{P_t}(M_s+Z_b{f}_{ma})-\frac{Z_b}{A_t}$

$2$