Introduction
Cylindrical pressure vessels are often used for a variety of functions. Some of these functions include the storage of liquefied petroleum gas, either during transit or in the countryside where cooking gas is not piped into their houses. Cylindrical pressure vessels are also used in the storage of fuel in gas stations in underground bankers, from where the fuel is pumped into the vehicles at the filling point. Due to these varied functions, it is important to understand the dynamics of stress and pressure on these cylindrical tanks. This is very important, especially in ensuring the structural integrity of these tanks so as to ensure the safe storage of the liquids and pressurized gases, and also the safety of the people and the environment around the cylindrical pressure vessels.
Most cylindrical pressure vessels are fitted with hemispherical ends. These cylindrical pressure vessels are made in such a way that they have uniform internal pressure. The material used in this case study is EN-18C (BS-540A40), also made of tempered and quenched steel alloys. This paper will present an analysis of the stresses in a cylindrical pressure vessel. The analysis of these stresses will employ the finite element method with the constant strain triangular element. For the purposes of this analysis it is assumed that the temperature in the cylindrical pressure vessel is constant and that h Poisson’s ratio is 0.30.
Stresses in cylindrical pressure vessels
Cylindrical pressure vessels experience different kinds of stresses. One of these kinds of stress is the circumferential stress, also known as hoop stress. This is a normal stress in the cylindrical pressure vessel that takes the pattern of the tangential or the azimuth direction. This is the force that is exerted circumferentially on the cylindrical pressure vessel. More precisely, circumferential stress is perpendicular to both the radius and the axis of the cylindrical pressure vessel. This force is exerted in both directions of all the particles of the walls of the cylindrical pressure vessel (Chandrupatla, 2004, Pp. 32). Hoop stress on a cylindrical pressure vessel is given by the following formula: σh=pd/2t
Where
σh = hoop stress
P = internal pressure of the cylindrical pressure vessel
D = internal diameter of the cylindrical pressure vessel
t = thickness of the walls of the cylindrical pressure vessel
(9.0 x 225 x 2)/ (2 x (240 – 225))
= 4050 / (2 x 15)
= 4050 / 30
= 135
Hoop stress for the cylindrical pressure vessel is 135 Mpa
Another kind of stress experienced by cylindrical pressure vessels is the axial stress, also known as longitudinal stress. As espoused earlier, most cylindrical pressure vessels have hemispherical ends fitted into them in order to have an enclosed space. When this is the case, internal pressure inside the cylindrical pressure vessel acts on the hemispherical ends thereby developing a force whose pattern pushes the hemispherical ends outside. This force is found along the axis of all cylindrical pressure vessels with hemispherical ends. With regards to the hoop stress as discussed above, the longitudinal or axial force is lesser when compared to the hoop stress experienced by the cylindrical pressure vessel (Banks, Nash, Spence & Conference on Pressure Equipment Technology: Theory and Practice. 2003). Axial stress in a cylindrical pressure vessel is given by the following formula.
σx=pd/4t
Where
σx = axial stress
P = internal pressure of the cylindrical pressure vessel
D = internal diameter of the cylindrical pressure vessel
t = thickness of the walls of the cylindrical pressure vessel
(9.0 x 225 x 2)/ (4 x (240 – 225))
= 4050 / (4 x 15)
= 4050 / 60
= 67.5
Axial stress for the cylindrical pressure vessel is 67.5 Mpa
Cylindrical pressure vessels experience yet another kind of stress. This is known as the radial stress. This is the kind of stress that a cylindrical pressure vessel experiences from its central axis. For a cylindrical pressure vessel with thick walls, the radial stress is opposite and equal in force to the gauge pressure experienced on the inside surface of the cylindrical pressure vessel and zero to the gauge pressure experienced on the outside surface of the cylindrical pressure vessel. The radial pressure of cylindrical pressure vessels is usually smaller when compared to the axial stress and circumferential stress of the pressure vessels. Because of this fact, there is a tendency to neglect the radial stress for cylindrical pressure vessels that have thin walls (Craig, 2011, Pp. 87). Radial stress is calculated using the following formula:
σr=-p/2
Where
σr = radial stress
P = internal pressure of the cylindrical pressure vessel
= -(9.0)/2
= -9.0/2
= -4.5
Radial stress for the cylindrical pressure vessel is -4.5 Mpa
Reflections
Analysis of the three stresses that affect cylindrical pressure vessels is very important when assessing the structural integrity of such vessels. As discussed earlier, cylindrical pressure vessels are used for the storage of different liquids and pressurized gasses. Some of these are very volatile and any leakages in the vessels can have adverse effects leading to losses and damages. The finite element method plays a very key role in this analysis. This particular exercise has brought to perspective the importance constant strain triangular element. The knowledge that stress on a cylindrical pressure vessel varies from one point to another along the shell profile of the cylindrical pressure vessel and through the girth of the vessel’s shell is in my domain.
As such, the analysis of the stress in cylindrical pressure vessels is important because fractures in these vessels are influenced by hoop stress when other external loads are absent. This is because hoop stress is the largest of the three stresses. Most sutures in these vessels occur from the inside where the hoop stress is greatest. Although the total strain experienced is the same on the inside and outside of the cylindrical pressure vessel because of its distribution over different circumferences, any fractures would theoretically start on the inside. In this regard, it is important to analyze and determine these elements in order to govern yielding by making the three stresses present in a manner that enhances the structural integrity of the cylindrical pressure vessel (Chandrupatla, 2004, Pp. 56).
Conclusion
The concept of constant strain triangular element is very important in the engineering field. This is because it is used in high stake situations where knowledge is used to prevent catastrophes. Pressurized gas that is stored in cylindrical pressure vessels exerts an outward pressure on the walls of the vessel. In order to ensure that accidents do not occur, it is important to understand the stress and strains that these vessels experience. As discussed earlier, some of these cylindrical pressure vessels are buried under the surface as storage bunkers for fuel. In order to determine the depth at which these vessels can be buried without negatively affecting their structural integrity, it is important to analyze the stress in cylindrical pressure vessels.
References
Banks, W. M., Nash, D. H., Spence, J., & Conference on Pressure Equipment Technology: Theory and Practice. (2003). Conference on Pressure Equipment Technology: Theory and Practice: 1-2 May 2003, the University of Strathclyde, Glasgow, UK. London [u.a.: Professional Engineering Publ.
Chandrupatla, T. R. (2004). Finite element analysis for engineering and technology. Hyderabad: Universities Press.
Craig, R. (2011). Mechanics of Materials. New York. John Wiley & Son.
