Example Nonlinear Programming Problems

Problem $1.($Difficulty level: $2)$

$(1)$ Derive the model for a uniform column of tubular section to carry a compressive load $P=2500k{g}_f$ for

minimum cost. The column is made up of a material that has a yield stress $({}_y)$ of $500k\frac{g_f}{c}{m}^2,$ modulus

of elasticity $($ E $)$ of $0\mathrm{.}85{10}^{6}k\frac{g}{c}{m}^2,$ and density $()$ of $0\mathrm{.}0025k\frac{g_f}{c}{m}^3.$ The length of the column is

$250$ cm $.$ The stress induced in the column should be less than the buckling stress as well as the yield

stress. The mean diameter of the column is restricted to lie between $2$ and $14$ cm $,$ and column with

thickness outside the range $0\mathrm{.}2$ to $0\mathrm{.}8$ cm are not available in the market. The cost of the column includes

material and construction costs and can be taken as $5W+2d,$ where $W$ is the weight in $k{g}_f$ and $d$ is the

mean diameter of the column in cm $.$

$(2)$ Design $d$ and $t$ to minimise the cost $($use at least two different optimisation techniques$/$software to find

the solution and compare the results$)$;

$(3)$ Discuss the effects of d and t on the cost, and the other important factors;

$(4)$ Based on the solution and discussion, prepare a report which provides detailed design guideline for the

column design $($assuming your colleagues will use this guideline in their designs of this type of columns$).$