Chapter 16, Problem 16P

Suppose that you are asked to design a column to support a compressive load P, as shown in Fig. P16.16a. The column has a cross-section shaped as a thin-wall edpipe as shown in Fig. P16.16b.

The design variables are the mean pipe diameter d and the wall thickness t. The cost of the pipe is computed by

$Cost=f\left(t,d\right)=c_{1}W+c_{2}d$

where $c_1=4\text{ and }{c}_2=2$ are cost factors and $W=$ weight of the pipe,

$W=\pi dtH\rho$

where $\rho =\text{density}$ of the pipe $\text{material=}0.0025kg/c{m}^3$. The column must support the load under compressive stress and not buckle. Therefore,

Actual stress ( $\rho$) $\le$ maximum compressive yield stress

$={\sigma }_y=550{\text{ kg/cm}}^{\text{2}}$

Actual stress $\le$ buckling stress

The actual stress is given by

$\sigma =\frac{P}{A}=\frac{P}{\pi dt}$

The buckling stress can be shown to be

${\sigma }_b=\frac{\pi EI}{H^{2}dt}$

where $E=\text{modulus}$ of elasticity and $I=\text{second}$ moment of the area of the cross section. Calculus can be used to show that

$I=\frac{\pi }{8}dt\left(d^2+t^2\right)$

Finally, diameters of available pipes are between $d_1$ and $d_2$ and thicknesses between $t_2$ and $t_2$. Develop and solve this problem by determining the values of d and t that minimize the cost. Note that $\begin{array}{l}H=275\text{ cm},P=2000kg,E=90,000\text{ kg}/{\text{cm}}^2,d_1=1\text{ cm},d_2=10\text{ cm},t_1=0.1\text{ cm},\\ \text{and }{t}_2=1\text{ cm}\end{array}$.

FIGURE P16.16

(a) A column supporting a compressive load P. (b) The column has a cross section shaped as a thin-walled pipe.