For the column cross section shown below $($a $18$in x $18$ in cross section, $12$ #$6$ bars. ties connecting in a fashion where $8$ bars are connected by a rectangular tie, $4$ similar ties are used.. $)$ with $12$#$6$ bars, assuming a clear cover of $1\mathrm{.}5$

inches to the ties, with a concrete strength of $4000$ psi $,$ and yield strength of all the reinforcement

of $60,000$psi:

a$)$ Select the required size and spacing of vertical ties. Are ties in addition to the perimetral one

required as shown in the figure below? $(1)$

b$)$ Develop the nominal and ultimate combinations of axial load and bending moment for the

strain conditions given below and plot the interaction diagram.

i$)$ Uniform compression strain over the depth of the section. $(1)$

ii$)$ Linear variation from maximum compression strain at the top of the section and zero

strain at the bottom. $(1)$

iii$)$ Balanced strain condition. $(2)$

iv$)$ Tension controlled limit strain distribution. $(1)$

v$)$ Determine if the section can support the following combination of loads $Pu=710$kips,

and $=50ft-$kips $(1)$

Plot side by side the $($Y$)$ nominal axial load vs $($X$)$ moment diagram with strain diagrams

corresponding to each of the points calculated, and the $(Y)$ nominal axial load vs $(x)$ curvature

diagram for the column section in Problem $1.$ Comment on the effect of axial load on nominal

moment and curvature. $(3)$