$1)$ The water tower shown schematically below is to be designed to best suit the needs of the client, Amigos de Jesus. Aluminum channel measuring $6"$ x $3\mathrm{.}5"$ x $0\mathrm{.}125"$ will be used for the columns and aluminum plate will be used for the tank. You will consider four design alternatives that make use of these materials, and ultimately make a recommendation to the client on the $$best$$ design that meets their current and future needs.

DETERMINE SECTION PROPERTIES OF A SINGLE CHANNEL $$ Start by determining the section properties of this channel cross$-$section. Specifically, determine:

$$ the area $($A$),$

$$ the location of the centroid x$,$ y$$

$$ the moment of inertia $($I$)$ about each of the centroidal axes

$$ the radius of gyration $($r$)$ about each of the centroidal axes

$2)$ DETERMINE SECTION PROPERTIES OF A TUBE CONSTRUCTED USING TWO CHANNEL SECTIONS $$ As an alternative cross$-$section, a tube consisting of two identical channels welded together along the entire length of the column may be used. Determine the section properties for this tube section. $($Specifically$,$ determine the same section properties that you did in the previous part for the channel alone.$)$ Note: It is suggested that you rely on principles of symmetry, answers from the previous part, and use of the Parallel Axis Theorem to determine these properties.

$3)$ CALCULATE ALLOWABLE AXIAL LOADS PER COLUMN FOR EACH OF FOUR DESIGN OPTIONS $$ Four design options are to be considered, as shown below. Relative to the standard design option $($Option #$1),$ modifications include the addition of midheight bracing for the weak direction $($Option #$2),$ the upgrading of foundation connections to provide fixity $($Option #$3),$ or the use of the tube $($double channel$)$ instead of the single channel $($Option #$4).$ For each option, determine the allowable axial load per column assuming that a factor of safety of $2\mathrm{.}5$ is to be provided against buckling or yielding. Use E $=10\mathrm{.}0$ x $10$ psi and y $=30,000$ psi.

$4)$CALCULATE APPROPRIATE TANK HEIGHT FOR EACH OF FOUR DESIGN OPTIONS $$ Based on available materials, the tank itself is to be constructed using $1/4$ in$.$ thick aluminum plate. The base and top of the tank are custom pieces that measure $8$ ft$.$ by $8$ ft$.$ Pieces for the tank sides are available in $1$ ft$.$ increments $($i$.$e$.$ measuring either $8$ ft$.$ by $1$ ft$.,8$ ft$.$ by $2$ ft$.,8$ ft$.$ by $3$ ft$.,$ etc.$).$ For each of the four options considered in the previous step, determine the maximum height of water that may be stored in the tank without exceeding the allowable axial load in the columns. Then, specify a tank height $($h$)$ by rounding up to the nearest ft$.$ dimension. $($Note: An overflow hole will be drilled at the maximum water height to prevent overloading the columns$).$ For simplicity, ignore the weight of the tank itself in all calculations; consider only the weight of the water. Note that water has a specific weight $=62\mathrm{.}4$ lb$/$ft$^3.$

$5)$ DETERMINE CONSTRUCTION COSTS FOR EACH OF FOUR DESIGN OPTIONS $$ Conduct a cost analysis to determine the relative costs of each option. Line item material and construction costs for each element are provided in the table below. Fill out the table to determine the total cost and unit cost $($$ per cubic ft$.$ of water stored$)$ of each option.

$6)$ MAKE A RECOMMENDATION TO THE CLIENT $$ In the space provided at the bottom of the page, provide a paragraph$-$long recommendation to the client on which of the four options they should consider implementing. Consider the following in making your recommendation: $$ Does the design meet the need of your client? Assume Amigos de Jesus currently houses $50$ children and $10$ adults. They would like to expand by adding $50$ more children in the near future. $$ Is the design economical? Funding is provided by donations from the United States through the hard work and dedication of the Amigos board. Funding also provides for building construction, meals, education, medical needs, and other critical items.