Problem $1$:

Part $1$: $($phase $2$ behavior$)$ For the section to the right:

$b=12\text{'}\text{'},d=22\text{'}\text{'},A_s=3\mathrm{.}61i{n}^2,$ grade $60$ steel and $f_c^{\text{'}}=5$ksi$(check{}_1).$

Find the phase $2$ neutral axis $($ x $)$ using the transformed area method

Part $2$: $($phase $3$ behavior @ failure$).$ Same section properties as Part $1$

a$.$ Draw the section and its strain and stress diagrams at failure

$0\mathrm{.}003).$ Use the ACI Whitney Stress Block. Label your diagrams with

the following symbols $a,c,d,{}_{c,}{}_t,T,C_c.$

b$.$ Compute the tensile and compressive forces at failure $($end of phase $3)$

c$.$ Determine the depth of the Whitney Stress Block $($a$)$ and the depth of the neutral axis $($c$).$

Compare the phase $3$ neutral axis $($c$)$ to the neutral axis $($x$)$ you found for phase $2$ in Part $1$

d$.$ Compute the moment arm and the moment capacity $(M_n)$

e$.$ Calculate the steel strain $({}_t),$ verify that the steel yields, and comment on whether the section

is usable by ACI code requirements on ${}_t$

f$.$ Determine $$ and calculate the usable $($design$)$ strength $M_n$

g$.$ Calculate the steel reinforcement ratio $=\frac{A_s}{b}d$ and check that it is within the min and max

limits given by ACI $($Table A$.7$ in your text book$).$ The max amount of steel limit for $$

corresponds to ${}_t=0\mathrm{.}004$

${}_1=\mathrm{.}85if{f}_c^{\text{'}}4,000$

$=\mathrm{.}85-(\mathrm{.}05)(f_c^{\text{'}}-4,000)\frac{1}{1,000}if4,000$