CE $648$ Homework #$3$

Question #$1$

A simply supported pretensioned pretopped double T$-$beam for a floor has a span of $70$ ft $(21\mathrm{.}3$ m $)$ and the geometrical dimensions shown in Figure P$1\mathrm{.}3.$ It is subjected to a gravity live$-$load intensity $W_L=480plf(7k\frac{N}{m}),$ and the prestressing tendon has an eccentricity at midspan of $e_c=19\mathrm{.}96in.(494$ mm $).$ Compute the concrete extreme fiber stresses in this beam at transfer and at service load, at the midspan section.

Section Properties

$A_c=,1185in.{?}^2$

$I_g=,109,621i{n}^4$

$C_b,=25\mathrm{.}65in.$

$C_t,=8\mathrm{.}35in.$

$S_b,=4274in.{?}^3$

$S_t,=13,128in.{?}^3$

$r^2,=\frac{1}{A_c}=92\mathrm{.}5in.{?}^2$

Notes:

The double T$-$beam is made of normal weight concrete. $f_c=6,000.$

$C_b=y_b($defined as the distance from the N$.$A$.$ to the bottom fiber$)$

$C_t=y_t($defined as the distance from the N$.$A$.$ to the top fiber$).$

Assume the use of $16-0\mathrm{.}5in.$ diameter, Grade $270$ low$-$relaxation strands.

Assume the effective prestressing stress in strands at prestress transfer $=189,000.$

Assume the effective prestressing stress in strands at service $=145,000.$