Thin walled pressure Vessel
Pressure vessels are closed structures that contain liquids or gases under pressure. Common
examples include tanks for compressed air, steam boilers, and pressurized water storage tanks.
Although pressure vessels exist in a variety of different shapes only thin-walled cylindrical and
1
spherical vessels are considered here. A vessel having a wall thickness less than about 10of inner
radius is called thin walled.
Cylindrical Vessels.
Consider the cylindrical vessel in figure below, have a wall thickness t, inner diameter d, and
subjected to gauge pressure p that developed within the vessel by a contained gas.
A small element of the vessel that is sufficiently removed from the ends and oriented is subjected
to normal stresses σ1 in the circumferential or hoop direction and σ2 in the longitudinal or axial
direction. The loadings are developed by the uniform hoop stress σ1 acting on the vessel’s wall,
and the pressure acting on the vertical face of the gas, we require;
ΣF =
0
𝜎1 𝐴𝜎 − 𝑃𝐴𝑃 = 0
𝜎1 (2𝐿𝑡) − 𝑃(𝑑𝐿) = 0
𝜎1 =
𝑃𝑑
2𝑡
(Stress in direction/circumferential/ hoop direction)
ΣF =
0
𝜎2 𝐴𝜎 − 𝑃𝐴𝑃 = 0
𝜎2 (𝜋𝑑𝑡) − 𝑃 �
𝜎2 =
𝑃𝑑
4𝑡
𝜋𝑑2
4
�=0
(stress in y direction/ longitudinal stress/ Axial stress)
σ1 , σ2 = normal stress in the hoop and longitudinal directions, respectively. Each is assumed to
be constant throughout the wall of the cylinder, and each subjects the material to
tension.
P = the internal gauge pressure the by the contained gas
d = the inner diameter of the cylinder
t = the thickness of wall (r/t ≥ 10)
By comparison, note that the hoop or stress is twice as large as the longitudinal or axial stress.
Consequently, when fabricating cylindrical pressure vessels from rolled-formed plates, the
longitudinal joints must be designed to carry twice as much stress as the circumferential.
Spherical Vessel
We can analyze a spherical pressure in similar manner. The vessel to have a wall thickness t,
inner radius r, and subjected to an internal gauge pressure p.
ΣF =
0
𝜎2 𝐴𝜎 − 𝑃𝐴𝑃 = 0
𝜎2 (𝜋𝑑𝑡) − 𝑃 �
𝜎2 =
𝑃𝑑
4𝑡
𝜋𝑑2
4
�=0
This is same results as that obtained for the longitudinal stress in the cylindrical pressure vessel.
Furthermore, from the analysis, this stress will be the same regardless of the orientation of the
hemispheric free-body diagram.
The above analysis indicates that an element of material taken from to either a cylinder or a
spherical pressure vessel is subjected biaxial stress, i.e, normal stress existing in only two
directions. The pressure also subjects the material to a radial stress σ3, which acts along the radial
line.
Example 1
Calculate the minimum wall thickness for a cylindrical vessel that is to carry a gas at a pressure
of 2 MPa. The diameter of the vessel is 1.25m, and the stress is limited to 45MPa.
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Example 2
A cylindrical pressure vessel has an inner diameter of 1.2m and a thickness ofo 12mm.
Determine the maximum internal pressure it can sustain that neither its circumferential nor its
longitudinal stress component exceeds 140MPa. Under the same conditions, what is the
maximum internal pressure that a similar size spherical vessel can sustain?
Example 3
A cylinder pressure vessel with an inside diameter of 1.50m is constructed by wrapping a 15mm
thick steel plate into a spiral and butt-welded the mating edges of the plate, as shown in figure
below . The butt-welded seams from as angle of 30o with a transverse plane through the cylinder.
Determine the normal stress σn perpendicular to the weld and the shearing stress τxy parallel to
weld when the internal pressure in the vessel is 1500KPa