Problem $5$ Application to a Design Problem $(5$ points$)$

Design a support column for a water tower where the height of the column is flxed. The basic tower design

is shown in the flgure below. The tank itself is a fixed load on the column, and the designer must choose the

radius of the column and the material from the provided options.

Assumptions:

The weight of the full tank acts as a flxed load vertically at Point B$.$

The support column has a circular cross$-$section.

The support column is subject to buckling failure.

Design Variables:

Column diameter: $d$

Material selection: $($Wood $($Spruce$),$ Concrete, Structural Steel, Cast Iron$)$

Variable Bounds:

Diameter of column: $0\mathrm{.}1d6\mathrm{.}0($meters$)$

Material Properties:

Numerical Constants:

Gravitational constant, $g=9\mathrm{.}8\frac{m}{s^2}$

Fixed water tank mass, $M_t=10,000kg$

Buckling factor of safety, $n=2$

Height of the support column, $h=12m.$

Column Buckling Formula:

To protect against buckling, we use the Euler column formula. The maximum load that can be supported

by a column with one end free and one end flxed is:

$P_{er}=\frac{{}^{2}EI}{4h^2}$

where $I=\frac{(\frac{d}{2}{)}^4}{4}$ for a circular cross$-$section.

Tosks:

$($i$)$ Calculate the maximum load that can be supported by the column for each material and diameter.

$($ii$)$ Find the minimum column diameter for each material such that the column supports the full weight

of the tank with a factor of safety $n=2.$

$($iii$)$ Plot the identifled pairs of material and minimum diameter in the consequence space $($i$.$e$.,$ cost vs$.$

maximum load$).$