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NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
UNIT-5
THIN CYLINDERS AND SPHERES
1. Derive the Expression for Hoop stress and longitudinal stress in a thin cylinder shells subjected to
internal fluid pressure? (OR) Show that in thin cylinder shells subjected to internal fluid pressure,
the Hoop stress is twice the longitudinal stress?
Ans:
THIN CYLINDRICAL VESSEL SUBJECTED TO INTERNAL PRESSURE:
The figure shows a thin cylindrical vessel in which a fluid under pressure is stored.
Let d = internal diameter of the thin cylinder
t = thickness of the wall of the cylinder
p = Internal pressure of the fluid
L = Length of the cylinder
EXPRESSION FOR CIRCUMFERENTIAL STRESS (HOOP STRESS):
Let p = Internal pressure of fluid
d = Internal diameter of the cylinder
t = Thickness of the wall of the cylinder
σ 1 = Circumferential or hoop stress in the material
NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
EXPRESSION FOR LONGITUDINAL STRESS:
Let p = Internal pressure of fluid, d = Internal diameter of the cylinder
t = Thickness of the wall of the cylinder, σ 2 = Longitudinal stress in the material
NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
2. A cylinder of internal diameter 2.5m and of thickness 5cm contains a gas. If the tensile stress in the
material is to exceed 80 N/mm2, determine the internal pressure of the gas?
Ans:
3. Derive the Expression for effect of internal pressure on the dimensions of a thin cylindrical shell?
Ans:
EXPRESSION FOR EFFECT OF INTERNAL PRESSURE ON THE DIMENSIONS OF A THIN
CYLINDRICAL SHELL:
&
NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
========================================================================
4. A cylindrical vessel is 1.5m diameter and 4m long is closed at ends by rigid plates. It is subjected to
an internal pressure of 3N/mm2. If the maximum principal stress is not to exceed 150 N/ mm2, find
the thickness of the shell. Assume E = 2 X 105 N /mm
2 and poisson’s ratio = 0.25. Find the changes
in diameter, length and volume of the shell?
Ans:
NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
&
5. Derive the Expression for Thin spherical shells subjected to internal fluid pressure?
Ans:
&
NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
The ratio of change in volume to the original volume is known as volumetric strain.
,
6. A vessel in the shape of a spherical shell of 1.20m internal diameter and 12mm shell thickness is
subjected to pressure of 1.6 N / mm2. Determine the stress induced in the material of the vessel?
Ans:
NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
UNIT-8
THICK CYLINDERS AND SPHERES
1. Derive the expressions for the stresses in a thick cylindrical shell subjected to an internal fluid
pressure? (OR) Derive the Lame’s equations?
Ans:
NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
These equations are called Lame’s Equations.
NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
Boundary conditions are :
2. Find the thickness of a metal necessary for a cylindrical shell of internal diameter 160 mm to
withstand an internal pressure of 8 N / mm2. The maximum hoop stress in the section is to exceed
35 N /mm2 ?
Ans:
NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
3. Determine the hoop stresses in a thick compound cylinder?
Ans:
4. A compound cylinder is made by shrinking a cylinder of external diameter 300mm and internal
diameter of 250mm over another cylinder of external diameter 250mm and internal diameter
200mm. the radial pressure at the junction after shrinking is 8 N/ mm2. Find the final stresses set up
NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
across the section, when the compound cylinder is subjected to an internal fluid pressure of 84.5 N/
mm2?
Ans:
NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
=========================================================================
NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
5. Derive the expression for the radial pressure and hoop stress for a thick spherical shell?
Ans:
NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
======================================================================
6. A thick spherical shell of 200mm internal diameter is subjected to an internal fluid pressure of 7 N /
mm2. If the permissible tensile stress in the shell material is 8 N / mm
2 , find the thickness of the
shell?
Ans:
NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
NRI INSTITUTE OF TECHNOLOGY STRENGTH OF MATERIALS-2 FOR CIVIL-2-2
M.VENKANNABABU
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