A post$-$tensioned rectangular concrete beam is to span $24\mathrm{.}0$ m $,$ simply supported and is to carry a superimposed UDL. The prestressing tendon consists of twenty$-$four $(24)12\mathrm{.}7mm$ diameter superstrands with an effective prestress level of ${}_{pc}=1250$MPa $($assumed constant along the profile$)$ after losses in a $96$ mm diameter duct. The concrete will have a characteristic compressive strength, $)=(0\mathrm{.}696.$ With a parabolic tendon profile, the eccentricity of the tendon from the centroidal axis at mid$-$span is $565$ mm $.$ The weight density of the concrete is $25k\frac{N}{m^3}.$

The nominal superimposed loads, as UDLs, are:

Permanent action $($Dead$),Wg=8\mathrm{.}2k\frac{N}{m}$ and $1$ mposed action $($Live$),Wq=26\mathrm{.}0k\frac{N}{m}.$

a$)$ Under full$-$service load, $G+Q,$ with the given level of effective prestress, ${}_{pe},$ has the beam cracked at mid$-$pan?

b$)$ At mid$-$span for Strength to the Ultimate Limit State $($U$.$L$.$S$.),$ is this cross$-$section adequate in bending for the loads to be carried? Your calculation of $o_{uo}$ is to be done from. The first principle is assuming the ERSE, not by some design approximation. As part of your calculation, what is the total strain in the prestressing tendon at mid$-$span, $pu?$ As part of your calculation, what is the effective depth at in$.$id$-$span, d$?$ As part of your calculation, what is the neutral axis parameter at ultimate, $k_u?$