Design a $12$ wide rectangular, Class U$,$ prestressed beam.

$$ Simply$-$supported, span length $=30$

$$ Material Properties:

o Concrete design strength: f $$c $=5000$ psi

o Concrete strength at prestress transfer: f $$ci $=4000$ psi

o Prestressing strands: $(1)/(2)$ diameter, $270$ ksi, low$-$relaxation

$$ Loads:

o Self$-$weight $(150$ lb$/$ft$3)$: Assume $12$ x $28$ section $$ WD $=350$ lb$/$ft

o Superimposed dead load $($sustained$)$: WSD $=650$ lb$/$ft

o Live load $($transient$)$: WL $=1250$ lb$/$ft

$$ Concrete stress limits: use ACI provisions $($with upper limit on ft for uncracked section$)$

$$ Prestress at transfer: fpi $=0\mathrm{.}70$ fpu

$$ Prestress loss: assume $20\%$

Use the following procedure:

$1.$ Find the minimum required section modulus, S$,$ then determine the required section depth.

$2.$ Find the required number of $(1)/(2)$ diameter strands so as not to exceed the stress limits $($initial

and after prestress loss$)$ for a section at mid$-$span. Assume straight strands with c$.$g$.$s

located at $3\mathrm{.}5$ to the bottom of the section.

$3.$ For the selected prestressing strands, check that stresses at the end section do not exceed

the stress limits