Mechanics of Materials II
UET, Taxila
Lecture No. (8&9)
Cylindrical vessel
with
hemispherical
ends
hemispherical ends

Consider now the vessel
shown in next Figure in
which the wall thickness of
the cylindrical and
hemispherical portions may
be different.
Cross-section of a thin cylinder with
hemispherical ends.
tc
ts

(this is sometimes
necessary since the hoop
stress in the cylinder is
twice that in a sphere of
the same radius and wall
thickness).
For the purpose of the
calculation the internal
diameter of both portions
is assumed equal.
From the preceding
sections the following
formula are known to
apply:

a- For the cylindrical portion

For the cylindrical portion hoop or
circumferential stress =
After substitution:
circumferential strain for cylindrical part
b- For the hemispherical ends
After substitution
Equilibrium

Thus equating the two strains in order
that there shall be no distortion of the
junction,
Then:

With the normally accepted value of
Poisson’s ratio for general steel work of
0.3, the thickness ratio becomes:
= 0.41

i.e. the thickness of the
cylinder walls must be
approximately 2.4 times
that of the hemispherical
ends for no distortion of the
junction to occur.

In these circumstances,
because of the reduced
wall thickness of the
ends, the maximum
stress will occur in the
ends.

For equal maximum
stresses in the two
portions the thickness of
the cylinder walls must be
twice that in the ends but
some distortion at the
junction will then occur.
Effects of end
plates and joints

The preceding sections
have all assumed uniform
material properties
throughout the components
and have neglected the
effects of endplates and
joints which are necessary
requirements for their
production.

In general, the strength of the
components will be reduced by
the presence of, for example,
riveted joints or welding, and
this should be taken into
account by the introduction of
a joint eficiency factor “  ”
into the equations previously
derived.

Welded joints are not as
strong as the parent plate
unless welds are
thoroughly inspected and,
if flawed, repaired during
manufacture - all of which
is expensive.
This strength reduction is
characterised by the:
weld or joint efficiency
η = joint strength / parent
strength


It varies from 100% for a
perfect weld (ie. virtually
seamless) through 7585% for a tolerably good
weld and much less than
that for bad weld.
Thus, for thin cylinders:

Hoop stress acting on
longitudinal Junction :
where L is the efficiency of the
longitudinal joints,
Why L with hoop stress and c with
longitudinal stress
longitudinal stress acting on
circumferential Junction :
where c is the efficiency of
the circumferential joints.
Pressure Vessels
For thin spheres:
Hoop stress acting on the Junction

Normally the joint
efficiency is stated in
percentage form and this
must be converted into
equivalent decimal form
before substitution into
the above equations.