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Show all calculations and steps clearly. Do not skip any step. The symbols have their usual meaning. Use ACI$-318.$ Assume a reasonable value of any parameter if needed.

Problem $1.$

A tied$-$column with square cross section supports an ultimate load of $P_u=400$kips and ultimate moment of $M_u=100$kip $-$ ft $.$ The column is in a braced frame and has an unsupported length of $9$ ft $.$ Assume $f_y=60$ksi and $f_c^{\text{'}}=4000.$

$($a$)$ Estimate the gross cross$-$sectional area and the size.

$($b$)$ Check whether the slenderness of the column can be neglected.

$($c$)$ Use the interaction diagram to find the reinforcement ratio and propose a bar size and number of bars. Assume the bars will be arranged in two faces. Identify which interaction diagram Fig A$6$a$,$ Fig A$-6$b$,$ etc. you used. Show all calculations.

Problem $2.$

A beam cross has effective depth of $20$ in and web width of $10$ in $.$ The factored ultimate shear force at the critical cross section near the support is $56$ kip.

$f_c^{\text{'}}=4500,f_{yt}=40,000.$

$($a$)$ Prove that shear reinforcement is required at the section.

$($b$)$ Calculate the spacing of No$.3$ double$-$legged stirrups.

$($c$)$ Check with the maximum spacing requirements.