Beam Cracking $(45$ points$).$ Consider the beam cross section shown below:

$A_s=3i{n}^2$

$f_y=60$ksi

$f_c=3$ksi

$f_r=7\mathrm{.}5\sqrt[\phantom{2}]{?}(f_c)=7\mathrm{.}5V(3000)$

$=0\mathrm{.}41$ksi

$E_s=29000$ksi

$E_c=57000\sqrt[\phantom{2}]{?}(f_c)=57000V(3000)$

$=3122$ksi

Note $-$ Follow the example given in class and questions $f)-$ i$)$ should be considerably easier if

you draw the diagrams of strain and stress through the depth of the beam.

a$)$ Find the $\frac{?}{\text{b}\text{a}\text{r}}(y)$ the centroid of the shape, measured relative to the top, accounting for the

difference in stiffness between the rebar and the concrete.

b$)$ Find and define $I_{UT}$ c$)$ What is the cracking moment $M_{CR}?$

d$)$ If the beam were simply supported and $20$ ft long $-$ what uniformly distributed load would cause that moment?

g$)$ What is the maximum compressive stress in the concrete when: $M=M_{CR}$ and $f_{ct}=f_{\text{r}\text{u}\text{p}\text{t}\text{u}\text{r}\text{e}}?$

Given the loading from part d$,$ where is the beam in compression $($diagram and sentence$)?$

h$)$ What is the stress in the steel under the same condition?i$)$ What is the strain in the steel?

i$)$ What is the strain in the steel?