Suppose that you are asked to design a column to support a compressive load P as shown
Question:
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.
Step by Step Answer:
Numerical Methods For Engineers
ISBN: 9780071244299
5th Edition
Authors: Steven C. Chapra, Raymond P. Canale