Tutorial 10:Pressure vessels
Solutions
1. A closed cylindrical steel vessel of radius 2000mm and
thickness 10mm has a yield stress of sY = 300MPa. It contains oil at a pressure of 950
kPa. Calculate the longitudinal and hoop
membrane stresses and deduce the load factor l
on membrane yield of the wall.
2. A cylindrical vessel is being designed with spherical end
closures. The cylinder radius is 3600mm
and it is to be made from steel with a yield stress of sY = 260MPa. The design pressure is 250 kPa and a load
factor of l = 1.8 is required against
first membrane yield according to von Mises.
Calculate the required thickness of the cylinder, and the required
thickness of the spherical ends. Note
that in reality, bending stresses will develop at the junction between the
cylinder and sphere, and these will cause first yield at a lower pressure than
the membrane theory calculation predicts.
Design pressure = 250 kPa and load factor =
1.8. So take the pressure at
failure as 250´1.8 = 450 kPa = 0.450 MPa
Cylinder
Sphere
Note that the spherical ends can be thinner
than the cylindrical body.
3. A large diameter shaft is made as a thin cylindrical
shell. It has a radius of 100mm and a wall
thickness 8mm. It carries a tensile
axial load of 930 kN and a torque of 40 kNm.
Calculate the membrane stresses in the cylinder wall, sketch the Mohr
circle, and deduce the principal stresses and their orientation to the
longitudinal axis. If the yield stress
is 380MPa, find the load factor l against failure
according to von Mises criterion. What
should the revised thickness be if the load factor must be increased to l = 2.0.
Sketch Mohr’s circle. Let
longitudinal direction be x and circumferential direction y.
Centre of circle at C = (sx + sy)/2 = (185.02 + 0)/2 =
92.51 MPa
Most tensile principal stress
s1 = C +
R = 92.51 + 122.03 MPa = 214.54 MPa
Most compressive principal stress
s2 = C -
R = 92.51 - 122.03 MPa = -29.52 MPa
Whilst this appears to be a clockwise rotation, the question did not
define whether the torque was clockwise or anticlockwise, so either + or – would
be OK as an answer.
4. A
short thin cylindrical aluminium tube with closed ends has a diameter of 600mm
and wall thickness 5mm. It is internally
pressurised to a pressure of 2.7MPa. It
carries an axial compressive load of 500 kN and a torque of 180 kNm. Calculate the membrane stresses in the
cylinder wall, sketch the Mohr circle, and deduce the principal stresses and
their orientation to the longitudinal axis.
If the yield stress is 230MPa, find the load factor against failure
according to von Mises criterion.
Cross-sectional area of tube:
A = 2pRt = 2p´300´5
= 9424.8 mm2
This is not a large enough safety margin
for most practical purposes
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