Related to Mechanics of Solid

1. JAHANGIRABAD INSTIUTE OF TECHNOLOGY
BARABANKI
Department of Mechanical Engineering
Thin Cylinders & Spheres
RAVI VISHWAKARMA
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 1

2. Stresses in thin Cylinders
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. In general three types of
stresses are developed in pressure cylinders viz.
circumferential or hoop stress, longitudinal stress in closed end
cylinders and radial stresses.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 2

3. The Thin-walled Pressure Vessel Theory
An important practical problem is that of a cylindrical or spherical
object which is subjected to an internal pressure p. Such a component
is called a pressure vessel, Fig.
Applications arise in many areas, for example, the study of cellular
organisms, arteries, aerosol cans, scuba-diving tanks and right up to
large-scale industrial containers of liquids and gases.
In many applications it is valid to assume that
the material is isotropic
the strains resulting from the pressures are small
the wall thickness t of the pressure vessel is much smaller than some
characteristic radius: t = ro-ri << ro,ri
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 3
p

4. Thin Walled Spheres
A thin-walled spherical shell is shown in Fig. Because of the symmetry
of the sphere and of the pressure loading, the circumferential(or
tangential or hoop) stress at any location and in any tangential
orientation must be the same (and there will be zero shear stresses).
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 4
σ t
σ t

5. Strain in the Thin-walled Sphere
The thin-walled pressure vessel expands when it is internally
pressurized. This results in three principal strains, the circumferential
strain σc (or tangential strain σ t ) in two perpendicular in-plane
directions, and the radial strain σ r. Referring to Fig., these strains are
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 5
p
C′
A′ B′
D′
A B
C
D

6. 10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 6
AB
ABBA
CD
CDDC
AC
ACCA
rc
−
=
−
=
−
=
''
,
''''
εε

7. Thin Walled Cylinder
The analysis of a thin-walled internally- pressurized cylindrical vessel
is similar to that of the spherical vessel. The main difference is that the
cylinder has three different principal stress values, the circumferential
stress, the radial stress, and the longitudinal stress σ l,
which acts in the direction of the cylinder axis, Fig.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 7
t
pr
l
2
=σ

8. Thick cylinders
Thick cylinders are the cylindrical vessels, containing fluid under
pressure and whose wall thickness is not small(t ≥ d/20).
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 8

9. Stresses in thick cylinders
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 axisymmetric about z-axis
and solving the differential equations of stress equilibrium in polar
co-ordinates.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 9

10. In general the stress equations of equilibrium without body forces can
be given as ---
For axisymmetric about z-axis
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 10
0=
∂
∂
θ
0
02
0
=+
∂
∂
+
∂
∂
=+
∂
∂
+
∂
∂
=
−
+
∂
∂
+
∂
∂
rzr
rzr
rzr
zrzzr
rzr
rrzr
τστ
τττ
σσσσ
θθθ
θ

11. Compound cylinders
An outer cylinder (jacket) with the internal diameter slightly smaller
than the outer diameter of the main cylinder is heated and fitted
onto the main cylinder. When the assembly cools down to room
temperature, a compound cylinder is obtained. In this process the main
cylinder is subjected to an external pressure leading to radial
compressive stresses at the interface (Pc) as shown in figure
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 11

12. Stresses due to Rotation
The bodies like rings, circular discs,cylinders,flywheel
,etc. invariably rotate at high speeds and due to
rotation they are subjected to large magnitudes of
centrifugal forces. The stresses caused by these forces
are distributed symmetrical about their axes of rotation.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 12