Mechanics of Materials II
UET, Taxila
Lecture No. (6)
Cylinders & Pressure vessels
Cylindrical or spherical
pressure vessels are
commonly used in
industry to carry both
liquids and gases under
pressure.
Classification of applications
Cylinders find many applications,
two of the most common
categories being :
a- fluid containers such as :
pressure vessels, hydraulic
cylinders, gun barrels, pipes,
boilers and tanks.
b- interference-fitted bearing
bushes, sleeves and the like.

Other applications

Cylinders can act as beams or
shafts eg. ( load building
blocks) but in the present
chapter cylinders are loaded
primarily by internal and
external pressures due to
adjacent fluids or to contacting
cylindrical surfaces.
Pressure Loading
When the pressure vessel is
exposed to this pressure,
the material comprising the
vessel is subjected to
pressure loading, and
hence stresses, from all
directions.
Factors that affect stresses
The normal stresses resulting from
this pressure are function of :
1- the radius of the element under
consideration,
2- the shape of the pressure vessel
(i.e., open ended cylinder, closed
end cylinder, or sphere)
3- the applied pressure.
Two types of analysis are
commonly applied to pressure
vessels.
The most common method is
based on a simple mechanics
approach and is applicable to
“thin wall” pressure vessels
which by definition have a ratio
of inner radius (r), to wall
thickness (t) of r/t ≥ 10.
The second method is based
on elasticity solution and is
always applicable
regardless of the r/t ratio
and can be referred to as
the solution for “thick wall”
pressure vessels.
Limiting proportions (approx)


Thin
d/t > 20
Thick
d/t < 20
t/d < 1/20
t/d > 1/20
t/d < 0.05
t/d > 0.05
Where d = Di = inner diameter
t = Cylinder thickness
Thin-Walled Pressure Assumptions
Several assumptions are made in this
method.
1) Plane sections remain plane
2) r/t ≥ 10 with t being uniform and
constant
3) The applied pressure, p, is the gauge
pressure (where p is the difference
between the absolute pressure and the
atmospheric pressure)
4) Material is linear-elastic,
isotropic and homogeneous.
 5) Stress distributions
throughout the wall thickness will
not vary
 6) Element of interest is remote
from the end of the cylinder and
other geometric discontinuities.
 7) Working fluid has negligible
weight.

THIN CYLINDERS AND SHELLS
1- THIN CYLINDERS
Thin cylinder representation
Classifications of Cylinders
Cylinders are classed as being
either :
 open - in which there is no
axial component of wall stress,
or
 closed - in which an axial
stress must exist to equilibrate
the fluid pressure.
Different types of open & closed Cylinders

When a thin-walled
cylinder is subjected to
internal pressure, three
mutually perpendicular
principal stresses will
be set up in the cylinder
material.
Types of stresses
Namely:
1- The circumferential
or hoop stress.
2- The longitudinal
stress.
3- The radial stress.

Provided that the ratio of
thickness to inside diameter
of the cylinder is less than
1/20, it is reasonably
accurate to assume that the
hoop and longitudinal
stresses are constant across
the wall thickness.

Also, the magnitude of the
radial stress set up is so
small in comparison with
the hoop and longitudinal
stresses that it can be
neglected.

This is obviously an
approximation since, in
practice, it will vary from
zero at the outside
surface to a value equal to
the internal pressure at
the inside surface.

For the purpose of the initial
derivation of stress formulae it
is also assumed that the ends
of the cylinder and any riveted
joints present have no effect on
the stresses produced; in
practice they will have an effect
and this will be discussed later.
Thin cylinders under internal
pressure
Hoop or circumferential stress
1- Hoop or circumferential stress

This is the stress which is
set up in resisting the
bursting effect of the applied
pressure and can be most
conveniently treated by
considering the equilibrium
of half of the cylinder.
It is required to
calculate the hoop
stress in terms of:
Pressure (p)
Inner diameter (d)
Thickness (t)

Half of a thin cylinder subjected to
internal pressure showing the hoop
and longitudinal stresses acting on
any element in the cylinder surface.

Consider the equilibrium of
forces in the x-direction
acting on the sectioned
cylinder shown in figure 2.
It is assumed that the
circumferential stress H
(or θ ( is constant through
the thickness of the
cylinder.
Figure (2)
Using the force
equilibrium to derive
an equation for hoop
stress
Calculating the total force owing to internal
pressure
 Total force on half-cylinder owing to
internal pressure
Resisting force owing to hoop stress

Total resisting force owing to hoop
stress
Force =
H set up in the cylinder walls=
Final form of hoop stress
Longitudinal stress
Longitudinal stress

Consider now the cylinder shown in
Next Figure.
Cross-section of a thin cylinder.
End Section of Cylindrical Thin-Walled Pressure
Vessel Showing Pressure and Internal Axial
Stresses
Using the force
equilibrium to derive
an equation for
longitudinal stress

Now consider the
equilibrium of forces
in the z-direction
acting on the part
cylinder shown in
next figure .
Force owing to internal pressure
Total force on the end of the cylinder
owing to internal pressure
Force on cylinder end :
Force =

For equilibrium of
forces we need to
calculate the End
section area
End Section Area
The cross-sectional area of the cylinder wall is
characterized by the product of its wall
thickness and the mean circumference
p D  t t
For the thin-wall pressure vessels where
D >> t
the cylindrical cross-section area may be
approximated by
pDt.
Longitudinal stress final form
Changes in
dimensions:
(a) Change in length

The change in length of
the cylinder may be
determined from the
longitudinal strain by
neglecting the radial
stress.
From Hooke’s Law
And change in length =
longitudinal strain x original
length
Then change in length =

(b) Change in diameter
As above, the change in
diameter may be
determined from the
strain on a diameter, i.e.
the diametral strain.
Now the change in diameter
may be found from a
consideration of the
circumferential change.
 The stress acting around a
circumference is the hoop or
circumferential stress H
giving rise to the circumferential
strain H.

Change in Diameter