Suppose that you are asked to design a column to support a compressive load P as shown in Figure. The column has a cross-section shaped as a thin-walled pipe as shown in Figure. The design variables are the mean pipe diameter d and the wall thickness t. The cost of the pipe is computed by

Cost = ?(t, d) = c₁ W + c₂d

Where c₁ = 4 and c₂ = 2 are cost factors and W = weight of the pipe,

W = ?dt Hp

Where p = density of the pipe material = 0.0025 kg/cm³. The column must support the load under compressive stress and not buckle. Therefore, Actual stress (?) ? maximum compressive yield stress

= ?_y = 550 kg/cm²

Actual stress ? buckling stress

The actual stress is given by

? = P/A = P/?dt

The buckling stress can be shown to be

?_b = ?El/H²dt

Where E = modulus of elasticity and l = second moment of the area of the cross section. Calculus can be used to show that

I = ?/8 dt(d² + t²)

Finally, diameters of available pipes are between d₁ and d₂ and thicknesses between t₁ and t₂. Develop and solve this problem by determining the values of d and t that minimize the cost. Note that H = 275 cm, P = 2000 kg, E = 900,000 kg/cm², d₁ = 1 cm, d₂ = 10 cm, t₁ = 0.1 cm, and t₂ = 1 cm.

H. (b) (a)