Consider the structural system shown below and design the pretensioned interior beam. As

shown in Fig. $1,$ the interior beam has a rectangular section, and is simply supported with a span

length of $30$ ft $($center to center of bearings$).$ The floor slabs that are simply supported by the

beams are made of $6$ in$.$ deep by $4$ ft wide hollow core slabs without toppings. The sectional

properties of the slabs are shown in Fig$\mathrm{.}2.$

The live loads on the floor is $100$ psf$.$ No additional dead loads other than the self$-$weight of slabs

and beams.

Material is selected by the engineer in consultation with the fabricator locally and the properties

are listed below: Concrete: f$$ci $=4000$ psi; f$$c$=5000$ psi, normal weight concrete; Prestressing steel:

$0\mathrm{.}5$ inch diameter, Grade $270,$ Low Relaxation $($Lolax$)$ strands; initial prestressing stress $=0\mathrm{.}75$fpu;

Mild reinforcing steel: Grade $60$ reinforcing bar. Consider a straight tendon profile.

$1)$ Calculate the live load, superimposed dead load, and self$-$weight of the interior beam. $(5$ pts$)$

$2)$ Conduct service level flexural design based on the midspan section. This member is to be

designed as a class U member. Use the PCI load table in Fig. $3$ to choose a preliminary section

geometry. Assume the maximum practical tendon eccentricity is at $3$ in$.$ from the bottom of the

section. Construct the feasible domain to select the least number of strands needed and

corresponding tendon eccentricity. If the tensile stress at prestress transfer exceeds the

allowable limit$,$ mild steel reinforcement may be added to compensate the tensile stresses,

however, all other allowable stress limits must be met. $(20$ pts$)$

$3)$ Estimate the prestress loss in the strands at midspan using a detailed procedure. How much is

that different from your assumption in part $2)?$ How will it impact your design? No need to redo

the design here. $(15$ pts$)$

$4)$ Calculate the transfer length of the strands. Check the allowable stress limit of the section at

the end of the transfer length. If this does not meet the allowable stress limits$,$ what could be

done to address this problem? No need to redo the design here. $(15$ pts$)$

$5)$ Check the section for minimum reinforcement requirements and ultimate strength capacity

under flexure. If any of those are not met, design the additional reinforcing steel needed. $(15$ pts$)$

$6)$ Design the shear reinforcement based on the most critical section only. $(20$ pts$)$

$7)$ Calculate the live load deflection and time$-$dependent deflection of the beam at erection and

in the long term. $(10$ pts$)$