A post$-$tensioned prestressed T$-$beam has the following details:

Span and loading:

simply supported $32m$ span

dead load comprises self$-$weight $($concrete density $25k\frac{N}{m^3})$ plus a nominal superimposed dead load $($SDL$)$ of $2k\frac{N}{m}$

nominal live load is $5k\frac{N}{m}$ and $30\%$ of the nominal live load is permanent

ultimate limit state load factors $[1\mathrm{.}2]$ for dead load and SDL$,$ and $[1\mathrm{.}5]$ for live load

single tendon with a parabolic profile and zero eccentricity at each end

all strands in the tendon are jacked simultaneously

the tendon is jacked from one end of the beam only

depth to centroid of tendon at midspan $1180mm$

midspan cross section

no passive reinforcement

Concrete:

$f_c^{\text{'}}=25$MPa at transfer and $32$MPa at $28$ days. Corresponding $E_c$ values are as given in Lecture $1.$

The beam will be located in an interior environment with an annual average temperature of $22C.$

Prestressing:

$15\mathrm{.}2mm$ diameter strands. $A_p=140m{m}^2$ per strand, $E_p=195,000$MPa,$f_{pb}=1790$MPa.

FOR THE MIDSPAN CROSS SECTION $($ONLY$)$:

Determine the effective prestressing force required using partial prestressing with the criteria that there shall be $0\mathrm{.}5$MPa of compression at the bottom of the beam under permanent load.

Calculate the minimum jacking force required assuming total losses of $20\%.$ Hence calculate the number of strands required if they are stressed to $80\%$ of their breaking load. Referring to the VSL Construction Products brochure page $9($available on Canvas$)$ select the appropriate tendon unit and the diameter of the zinc$-$coated circular metal duct $($metal sheathing$)$ for this number of strands.

Using the number of strands calculated in Question $2,$ recalculate the jacking force, and then calculate all losses. Use a value of the friction coefficient $$ that is the bottom $($lowest$)$ of the recommended range for metal sheathing, and use a value of the wobble coefficient ${}_p$ that is at the bottom $($lowest$)$ of the recommended range. $6mm$ anchor draw$-$in$.$ Use a basic relaxation value $R_b$ of $3\mathrm{.}5\%.$ In a table, show the loss of tendon force $($in both $kN$ and as a percentage of the jacking force$)$ for all six possible components of the loss of tendon force, and compare the total loss to the initial assumption of $20\%.$ Comment on any difference $($but do not adjust the jacking force and repeat the calculation$).$

Calculate the ultimate moment capacity of the prestressed beam using the approximate method from AS $3600(0\mathrm{.}85M_u)$ and compare this with the design moment $M^{**}.$ Does the cross section have adequate ultimate bending capacity?