Prove that hoop stress is twice the longitudinal stress in a cylindrical pressure vessel.

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)

a) cold, dry

explanation:

continental polar masses are cold and dry air masses because the cold air cannot hold much moisture. as they move into warmer air, they can trigger precipitation.

when it gains or loses electrons