THICK CYCLES

1. MODERN INSTITUTE OF
ENGINEERING & TECHNOLOGY
MECHANICAL
DEPARTMENT
5th SEMESTER
THICK CYLINDERS SUBJECTED TO
INTERNAL AND EXTERNAL PRESSURE
MADE BY:-SUKANTA MANDAL

2. When a thick-walled tube or cylinder
is subjected to internal and external
pressure a hoop and longitudinal
stress are produced in the wall.
Subject: Radial and tangential stress in thick-walled
cylinders or tubes with closed ends - with internal and
external pressure

3. INTRODUCTION
The thickness of the cylinder is large compared to that of thin cylinder.
i. e., in case of thick cylinders, the metal thickness ‘t’ is more than ‘d/20’,
where ‘d’ is the internal diameter of the cylinder.
Magnitude of radial stress (pr) is large and hence it cannot be neglected. The
circumferential stress is also not uniform across the cylinder wall. The radial
stress is compressive in nature and circumferential and longitudinal stresses are
tensile in nature. Radial stress and circumferential stresses are computed by
using ‘Lame’s equations’.

4. LAME’S EQUATIONS (Theory) :
A
S
S
U
M
P
T
I
O
N
S
:
1. Plane sections of the cylinder normal to its axis remain plane and normal even
under pressure.
2. Longitudinal stress (σL) and longitudinal strain (εL) remain constant throughout the
thickness of the wall.
3. Since longitudinal stress (σL) and longitudinal strain (εL) are constant, it follows that
the difference in the magnitude of hoop stress and radial stress (pr) at any point on
the cylinder wall is a constant.
4. The material is homogeneous, isotropic and obeys Hooke’s law. (The stresses are
within proportionality limit).

5. LAME’S EQUATIONS FOR RADIAL PRESSURE AND
CIRCUMFERENTIAL STRESS
Consider a thick cylinder of external radius r1 and internal radius
r2, containing a fluid under pressure ‘p’ as shown in the fig. Let ‘L’ be the
length of the cylinder.
p
r2
r1
p

6. Stress in Axial Direction
The stress in axial direction at a point in the tube or cylinder
wall can be expressed as:
σa = (pi ri
2 - po ro
2 )/(ro
2 - ri
2) (1)
where
σa = stress in axial direction (MPa, psi)
pi = internal pressure in the tube or cylinder (MPa, psi)
po = external pressure in the tube or cylinder (MPa, psi)
ri = internal radius of tube or cylinder (mm, in)
ro = external radius of tube or cylinder (mm, in)

7. Stress in Axial Direction
Stress in Axial Direction
The stress in axial direction at a point in the tube or cylinder wall
can be expressed as:
σa = (pi ri
2 - po ro
2 )/(ro
2 - ri
2) (1)
where
σa = stress in axial direction (MPa, psi)
pi = internal pressure in the tube or cylinder (MPa, psi)
po = external pressure in the tube or cylinder (MPa, psi)
ri = internal radius of tube or cylinder (mm, in)
ro = external radius of tube or cylinder (mm, in)

8. The stress in circumferential direction - hoop stress - at
a point in the tube or cylinder wall can be expressed
as:
σc = [(pi ri
2 - po ro
2) / (ro
2 - ri
2)] - [ri
2 ro
2 (po - pi) / (r2 (ro
2 - ri
2))] (2)
where
σc = stress in circumferential direction (MPa, psi)
r = radius to point in tube or cylinder wall (mm, in) (ri < r
< ro)
maximum stress when r = ri (inside pipe or cylinder)
Stress in Circumferential
Direction - Hoop Stress

9. Resultant Stress
Resultant Stress
Combined stress in a single point in the
cylinder wall cannot be described by a single
vector using vector addition. Instead stress
tensors (matrixes) describing the linear
connection between two physical vectors
quantities can be used.

10. Stress in Radial Direction
Stress in Radial Direction
The stress in radial direction at a point in the tube or cylinder
wall can be expressed as:
σr = [(pi ri
2 - po ro
2) / (ro
2 - ri
2)] + [ri
2 ro
2 (po - pi) / (r2 (ro
2 - ri
2))] (3)
maximum stress when r = ro (outside pipe or cylinder)

11. Example
Example - Stress in Thick walled Cylinder
In a cylinder with inside diameter 200 mm (radius 100 mm) and outside
diameter 400 mm (radius 200 mm) there is a pressure 100 MPa relative to
the outside pressure.
Stress in axial direction can be calculated as
σa = (((100 MPa) (100 mm)2 - (0 MPa) (200 mm)2) / ((200 mm)2 - (100
mm)2)
= 33.3 MPa
Stress in circumferential direction - hoop stress - at the inside wall (100
mm) can be calculated as
σc = [((100 MPa) (100 mm)2 - (0 MPa) (200 mm)2) / ((200 mm)2 - (100
mm)2)] - [(200 mm)2 (100 mm)2 ((0 MPa)- (100 MPa)) / ((100 mm)2 ((200
mm)2 - (100 mm)2))]
= 167 MPa
Stress in radial direction at the inside wall (100 mm) can be calculated as
σr = [((100 MPa) (100 mm)2 - (0 MPa) (200 mm)2) / ((200 mm)2 - (100
mm)2)] + [(200 mm)2 (100 mm)2 ((0 MPa)- (100 MPa)) / ((100 mm)2 ((200
mm)2 - (100 mm)2))]
= -100 MPa

12. 1. Variations of Hoop stress and Radial stress are parabolic across the cylinder
wall.
2. At the inner edge, the stresses are maximum.
3. The value of ‘Permissible or Maximum Hoop Stress’ is to be considered on the
inner edge.
4. The maximum shear stress (σ max) and Hoop, Longitudinal and radial strains
(εc, εL, εr) are calculated as in thin cylinder but separately for inner and outer
edges.
IMPORTANT POINTS
DJ996

13. THANK YOU