Stress in thin cylinder - Hoop stress and Longitudinal stress.

Cylindrical and spherical vessels are used in the engineering field to store and transport fluids. Such vessels are tanks, boilers, compressed air receivers, pipelines, etc. These vessels, when empty, are subjected to atmospheric pressure internally as well as externally and the resultant pressure on the walls of the shell is nil. But whenever a vessel is subjected to internal pressure(due to air, water, steam, etc.) its walls are subjected to tensile stresses

THIN CYLINDERS

A cylinder is considered thin when the ratio of its inner diameter to the wall thickness is more than 15.

t/d <= d/10 to d/15 , it is called thin cylindrical shell.

t = thickness of the shell,

d =internal diameter of the shell.

Boiler shells, pipes, tubes, and storage tanks are treated as a thin cylinder.

In a thin cylindrical shell, hoops stress and longitudinal stresses are constant over the thickness and radial stresses are negligible.

Stresses in thin cylindrical shells

whenever, a thin cylindrical shell is subjected to internal pressure (p). Its walls are subjected to two types of tensile stresses.

(a) Hoop stress (circumferential stress)

(b) Longitudinal stress.

Hoop stress (circumferential stress)

The hoop stress is the force exerted circumferentially (perpendicular both to the axis and to the radius of the object) in both directions on every particle in the cylinder wall

Consider a thin cylinder subjected to internal pressure as shown in fig.

P = internal pressure

σc = circumferential stress in the shell material

d = internal diameter of shell

t =thickness of shell

Total pressure = force*area

= p*d*l

Resisting area A= 2*t*l

σc = P/A

= pdl/ 2tl

= pd / 2t

Longitudinal stress

Longitudinal stress is the stress in a pipe wall, acting along the longitudinal axis of the pipe. And it is produced by the pressure of the fluid in the pipe.

Consider a thin cylinder subjected to internal pressure as shown in fig.

P= internal pressure

σl = circumferential stress in the shell material

d= internal diameter of shell

t=thickness of shell

Total pressure = force*area

=(p*π*d^2)/4

Resisting area A= πdt

σl = P / A

= pd / 4t

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