Problem $1$

A two$-$span prestressed concret$$ beam has the

cross$-$section shown in Figure $1$a$.$ Section properties are:

Cross$-$sectional area $A_g=360,000m{m}^2$

Depth to centroid $y_O=426mm$

Second moment of area $I_g=50\mathrm{.}89{10}^{9}m{m}^4$

The beam has a prestressing cable with a parabolic profile

in both spans as shown in Figure $1$a$.$ The prestressing force

$P_g$ is $1600$ kN $,$ assumed constant throughout the beam.

The beam is designed to carry a live load of $20k\frac{N}{m}$ in

addition to its own weight.

Material properties are:

Concrete: $,f_o^{\text{'}}=40$MPa,$=0\mathrm{.}766,E_c=32,000$MPa

Prestressing steel: $A_p=1600m{m}^2,f_{py}=1564$MPa,$E_p=190,000$MPa

a$)$ Determine the secondary moments and the reactions due to prestress. Draw the

secondary moment diagram and show the directions of the reactions.

b$)$ For the section at the interior support B$,$ calculate the flexural moment capacity and

check whether it conforms to AS$3600$ requirements. Design any additional

reinforcement needed.

c$)$ For the midspan section D$,$ calculate the strength at transfer $P_u$ and determine whether

or not it satisfies AS$3600$ requirements. If it does not, state briefly what you would do

to fix the problem $($no calculations required$).$

The maximum prestressing force during jacking is $2100$ kN and the concrete strength

is $32$ MPa $.$ solve properly