discover the maximum safe air pressurea

Question

Discover the maximum safe air pressure:

A cylindrical compressed air drum is equal to 2 m in diameter along plates 12.5 mm thick. The efficiencies of the longitudinal (η_l) and circumferential (η_c) joints are equal to 85% & 45% respectively. If in the plating the tensile stress is to be limited to 100 MN/m², discover the maximum safe air pressure.

Solution

The efficiency of the joint effect the stresses induced. For a seamless shell (with no joints), efficiency is equal to 100%. While the efficiency of joint is less than 100%, the stresses are enhanced accordingly.

Therefore, if η is the efficiency of a joint in the longitudinal direction, affecting the hoop stress, then the stress shall be given as,

σ _h  =    pd/ (4t ×η_I)

Here, the diameter d = 2 m = 2000 mm.

Thickness, t = 12.5 mm.

Limiting tensile stress = 100 MN/m² = 100 N/mm².

Letting the circumferential joint which influences the longitudinal stress,

Pd/ (4t × ηl) = 100

p × 2000 / (4 × 12.5 × 0.45) = 100

 p = 1.125 N/mm²

Likewise, considering the longitudinal joint which influences the hoop stresses,

pd / (2t × ηc) = 100

p × 2000 / (2 × 12.5 × 0.85) = 100

p = 1.063 N/mm²

Evidently, safe pressure is governed through hoop stress.

Therefore, maximum safe air pressure = 1.063 N/mm².