In South African gold mines, cables are used to lower worker cages down mine sha

Question

In South African gold mines, cables are used to lower worker cages down mine shafts. Ordinarily, the cables are made of steel. To save weight, an engineer decides to use cables made of aluminum. A design requirement is that the stress in the cable resulting from the self-weight must not exceed one tenth of the ultimate stress of the material (0.1 sigma u). Use the following weight densities = 77 kN/m3 and gamma aluminum = 27.1 kN/m3 Use the following ultimate strengths sigma u-steel = 800 MPa and sigma u-aluminum = 470 MPa. Use the following moduli E steel = 200 GPa and E aluminum = 73 GPa. Assuming that the cross-sectional area of each cable is the same at 800 mm2, determine the elongation of each cable when the maximum stress in the cable is 0.1 sigma u-(remember that the max stress will be at the point where the full weight of the cable is supported)

Explanation / Answer

u-steel = 800Mpa                         1Gpa = 1000 Mpa

u-alum = 470 Mpa

Esteel = 200 Gpa                           1pa = 1 N/m^2

Ealum = 73Gpa

A = 800 mm^2 = 8X10^(-4) m^2

steel = 77 kN/m^3 =>

alum = 27.1 kN/m^3 =>

length of steel cable l-steel = 0.1*800*10^4/(77*8) = 1298.7 m

similarly

length of aluminum cable = 2167.8 m (conforming to = 0.1*u)

= /E

=>delta-L = *L

=> elongation of steel cable = * L/E

                     = 0.1*800*1298.7*10^6/(200*10^9)

                     = 0.519 m

similarly elongation of aluminium cable = 1.395 m

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