1. G H Patel College of Engineering And Technology 2151907 - Design of Machine Elements
cylindrical pressure vessel subjected to internal and external pressure
Roshan Qureshi ( )
Hardik Raiyani ( )
Kuldeep Raj ( )
Jigar Rana ( )
Dhruv Sakaria ( )
Sohel Shaikh ( )
Vinay Sudani ( )

2. Thin cylinder :-
If the wall thickness is less than about 7% of the inner diameter then the cylinder may be treated as a thin one.
Thin walled cylinders are used as boiler shells, pressure tanks, pipes and in other low pressure processing equipment's.
A thin cylinder is also defined as one in which the thickness of the metal is less than 1/20 of the diameter of the cylinder. In thin cylinders, it can be assumed that the variation of stress within the metal is negligible, and that the mean diameter, dm is approximately equal to the internal diameter, di.
In general three types of stresses are developed in pressure cylinders:-

3. When a thin cylinder is subjected to internal pressure,
three mutually perpendicular principle stresses will be
setup in the cylinder material.

4. circumferential or hoop stress.
longitudinal stress in closed end cylinders.
radial stresses.
Radial stress in thin cylindrical shells can be neglected as the radial pressure is not generally high and that the radial pressure acts on a larger area.
The internal pressure, p tends to increase the diameter of the cylinder and this produces a hoop or circumferential stress (tensile).
If the stress becomes excessive, failure in the form of a longitudinal burst would occur.

5. Circumferential or Hoop stress :-
Hoop stress t1
Projected area

7. Longitudinal stresses :-
Longitudinal stress, t2
Pressure area
Internal pressure, p

9. Thick Cylindrical Shells Subjected to an Internal Pressure :-
For thick cylinders such as guns, pipes to hydraulic presses, high pressure hydraulic pipes the wall thickness is relatively large and the stress variation across the thickness is also significant. In this situation the approach made in the previous section is not suitable. The problem may be solved by considering an axisymmetry about z-axis and solving the differential equations of stress equilibrium in polar co-ordinates. In general the stress equations of equilibrium without body forces can be given as

10. For axisymmetry about z-axis ∂/∂θ = 0 and this gives

11. This can be written in the following form:
If we consider a general case with body forces such as centrifugal forces in the case of a rotating cylinder or disc then the equations reduce to

12. It is convenient to solve the general equation so that a variety of problems may be solved. Now as shown in figure on next slide, the strains εr and εθ may be given by

13. Representation of radial and circumferential strain.

14. Combining previous equations we have
& previous equation we have
For a non-rotating thick cylinder with internal and external pressures pi and po we substitute ω = 0 and gives
A typical case is shown in figure in the next slide. A standard solution for equation is σr = c rn where c and n are constants. Substituting this in equation and also combining with
where c1 and c2 are constants.

15. A thick cylinder with both external and internal pressure

16. Boundary conditions for a thick cylinder with internal and external pressures pi and po respectively are:
at r = ri σr = -pi
and at r = ro σr = -po
The negative signs appear due to the compressive nature of the pressures. This gives

17. The radial stress σr and circumferential stress σθ are now given by
It is important to remember that if σθ works out to be positive, it is tensile and if it is negative, it is compressive whereas σr is always compressive irrespective of its sign.

18. Stress distributions for different conditions may be obtained by simply substituting the relevant values in equation. For example, if po = 0 i.e. there is no external pressure the radial and circumferential stress reduce to

19. The stress distribution within the cylinder wall is shown in figure
Radial and circumferential stress distribution within the cylinder wall when only internal pressure acts.

20. In the design of thick cylindrical shells, the following equations are mostly use:
Lame’s equation
Birnie’s equation
Clavarino equation
Barlow’s equation
The use of these equations depends upon the type of the material used and the end construction.

21. Lame’s equation :-

22. Birnie’s equation :-

23. Clavarino‘s equation :-

24. Barlow’s equation :-

25. Thick Cylindrical Shells Subjected to an external Pressure :-
In case of thick-walled pressurized cylinders, the radial stress, r , cannot be neglected.
Assumption – longitudinal elongation is constant around the plane of cross section, there is very little warping of the cross section, εl = constant
l = length of cylinder
 F = 0
2(θ)(dr)(l) + r (2rl) – (r + dr) [2(r + dr)l] = 0 dr θ
r + dr r
(dr) (dr) is very small compared to other terms ≈ 0
θ – r – r dr
dr
= 0
(1)

26. εl = – μ θ r εl θ + r θ – r – r r + r + r r r 2 r θ μ dr
Deformation in the longitudinal direction θ
+ r
= 2C1 = εl E μ
constant
(2)
εl = – μ θ E r
– μ
θ – r – r dr
dr
= 0
(1)
Subtract equation (1) from (2),
r + r + r dr
dr
= 2C1
Consider,
d (r r 2) dr =
r 2
dr
+ 2r r
2rr + r2 dr
dr
= 2rC1
Multiply the above equation by r
d (r r 2) dr =
2rC1
r r 2
= r2C1
+ C2 r
= C1 C2 r2 + θ –

27. θ r l r r pi ri2 - po ro2 – ri2 ro2 (po – pi) / r2
= - pi at r = ri
Boundary conditions r
= - po at r = ro θ =
pi ri2 - po ro2 – ri2 ro2 (po – pi) / r2
ro2 - ri2
Hoop stress r =
pi ri2 - po ro2 + ri2 ro2 (po – pi) / r2
ro2 - ri2
Radial stress
pi ri2 - po ro2 l =
ro2 - ri2
Longitudinal stress

28. Special case, po (external pressure) = 0
ro2 - ri2
(1 -
ro2 r2 )
pi ri2 θ =
pi ri2
ro2 - ri2
(1 +
ro2 r2 )
Hoop stress distribution, maximum at the inner surface
Radial stress distribution, maximum at the inner surface