Answer:
Proof in explanation.
Explanation:
Consider a thin cylinder, whose thickness to diameter ration is less than 1/20, the hoop stress can be derived as follows:
Let,
L = length of cylinder
d = internal diameter of cylinder
t = thickness of wall of cylinder
P = internal pressure
σH = Hoop Stress
σL = Longitudinal Stress
Total force on half-cylinder owing to internal pressure = P x Projected Area = P x dL (Refer fig 9.1)
Total resisting force owing to hoop stress setup in walls = 2 σH L t
Therefore,
P d L = 2 σH L t
σH = Pd/2t _____________ eqn (1)
Now, for longitudinal stress:
Total force on end of cylinder owing to internal pressure = P x Projected Area = P x πd²/4
Area resisting this force = π d t (Refer fig 9.2)
Longitudinal Stress = Force/Area
σL = (Pπd²/4)/(πdt)
σL = Pd/4t ____________ eqn (2)
Dividing eqn (1) by eqn (2)
σH/σL = 2
σH = 2 σL (Hence, Proved)
