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First Serbian (26th YU) Congress on Theoretical and Applied Mechanics

Kopaonik, Serbia, April 10-13, 2007

ANALYTICAL AND NUMERICAL SOLUTIONS OF INTERNAL FORCES BY CYLINDRICAL PRESSURE VESSEL WITH SEMI-ELLIPTICAL HEADS

P. Baličević1, D. Kozak2, D. Kraljević1

1 Josip Juraj Strossmayer University of Osijek,

Faculty of Agriculture, Trg Sv. Trojstva 3, HR-31000 Osijek, Croatia

e-mail: pbalicevic@pfos.hr, dkraljevic@pfos.hr

2 Josip Juraj Strossmayer University of Osijek,

Mechanical Engineering Faculty, Trg I. Brlić-Mažuranić 2, HR-35000 Slavonski Brod, Croatia

e-mail: dkozak@sfsb.hr

Abstract:

Increasing needs for pressure vessels exploitation in the refineries, chemical plants, power plants and other processing facilities requires contemporary design and accurate analysis of the shell in the design phases. In this paper the solutions for internal forces and displacements in the thin-walled cylindrical pressure vessel with ellipsoidal heads using the general theory of thin-walled shells of revolution have been proposed. Expressions which describe the distribution of the forces and displacements in the thin-walled shell are given in mathematical form. The distribution of internal forces is illustrated also by diagrams for selected design of vessel with defined geometrical characteristics. The accuracy of the analytical solutions has been proved by finite element analysis.

Key words: thin-walled shell, cylindrical pressure vessel, semi-elliptical heads, internal forces

1. Introduction

Modern forming technology of the thin-walled steel elements enables heads fabrication for cylindrical pressure vessels of the favourable construction form. Cylindrical pressure vessels with semi-ellipsoidal or toroidal-sphere heads are more often installed in any application area in the industrial plant systems. Their practical advantage over the vessels with hemisphere heads is reflected in total pressure vessel length decrease without significant reducing of useful internal volume.

A pre-condition of an adequate shaping pressure vessel design, as well as selection of suitable material and technological operations during the vessel fabrication is knowing of real stress distribution in the vessel parts. Shell theory solutions established by famous researchers (Lourye 1947, Goldenveiser 1953, Novozhilov 1962 and others) can be directly used for stress computation in thin-walled pressure vessels of simple geometry. These analytical solutions for internal forces, moments and displacements were studied and used in this paper.

Direct application of shell theory solutions encounters series of mathematical difficulties with pressure vessels fabricated of more shells of diverse geometric figures, especially if complex contours are taken into account. Such example is a cylindrical pressure vessel with heads in the form of revolution ellipsoid half. Wall curvature radii are not constant values with semi-ellipsoidal heads being an additional problem for further transformation of shell theories general solutions. Aiming to develop stress calculation method while constructing these vessels, this paper deals with expressions for membrane forces and displacements in semi-ellipsoidal head shells. Understanding these solutions is vital for conducting final vessel calculation by the general shells theory intended to be further studied by the authors. Conducting of internal force analysis applying general theory requires setting and solving of boundary conditions equations. These conditions can be set out if we understand displacement distribution of certain membrane theory.

2. Membrane forces in the pressure vessel walls

The medium flat of the semi-ellipsoidal head occurs by ellipse arc rotation around its minor axes. The main curvature radii of such head are unbroken meridian angle functions. Known from the differential geometry [1], they can be set out as follows: hemisphere or toroidal sphere heads. The sphere segment-like heads are seldom applied since the wall bending, accompanied by high value stresses, occurs on the spot of their joint with cylindrical part. However, stress caused by bending on the points of transferring sphere part into toroidal one also occurs with toroidal-sphere heads. The bending can be considerably reduced if heads in the form of revolution ellipsoid half are used instead of toroidal–sphere ones. It is possible since both of them are geometrically similar to each other.

The mid-plane of the semi-ellipsoidal head occurs by ellipse arc rotation around its minor axes. The main curvature radii of such head are continuing meridian angle functions. Known from the differential geometry [1], they can be set out as follows:

[pic], (1)

[pic], (2)

where [pic] is a parameter determining ellipse form, [pic] ellipsoid curvature radius in the head vertex, a – ellipse major semi-axes, b – minor semi-axes, [pic] - meridian angle. The basic measures of cylindrical pressure vessels, closed with heads in the form of revolution ellipsoid half, can be seen in Figure 1. A major ellipsoid semi axis is equal to cylindrical part radius (a = R), and a minor one equals to the head height (b = H). Radius and head height ratio is usually applied as 2 in the praxis (R/H = 2). Such a head construction is characterized by the ellipsoid parameters γ = 3 and R0 = 2R, where γ is parameter which defines the form of ellipse, R0 is the roundness ratio in the corner of the ellipsoid.

[pic]

Fig. 1. Geometry of the cylindrical pressure vessel with ellipsoidal heads

Meridian and circular normal forces in thin-walled shells of revolution by the membrane theory [2] is as follows:

[pic], (3)

[pic], (4)

where [pic] is the total sum of all external forces affecting vessel shell to the monitored section, r is mid-plane radius in considered point of the wall, [pic]is component of the continuous wall loading in the external normal direction, [pic] meridian normal wall force and [pic]is circular normal force in the wall.

Upper index [pic] denotes sizes pertaining to a membrane stress condition. Corresponding boundary condition sizes will be denoted by the superscript [pic], and sizes without superscript refer to total solutions.

If the vessel is exposed only to medium internal pressure in the vessel, the resultant of the shell load is [pic]. Substitution of the value into (3) and taking [pic] result in:

[pic], (5)

Substitution of (5) into (4) alongside continuous wall loading in the normal direction value [pic] will result in (pressure load) circular force formula:

[pic], (6)

Formulas (5) and (6) are generally valuable for any part of the pressure vessel wall. They indicate that final formulas for internal forces depend on the law of the main curvature radii change of the vessel wall. In the further text internal forces in the head walls are denoted by a hyphen whereas cylindrical part forces with two hyphens. Substitution of the curvature radius into (5) and (6) with the formulas (1) and (2) results in formulas for meridian and circular forces into ellipsoidal head:

[pic], (7)

[pic], (8)

Formulas (7) and (8) can be set out by diagrams in the dimensionless form. That’s why a series of points has been selected on the head meridian with meridian angle increment [pic]. Since these points do not divide ellipse arc into even parts, a few more points (denoted by asterisk in the table) have been added between them. Membrane internal forces values were computed, in selected points, by the formulas (7) and (8) for chosen parameter value [pic] and presented in Table 1. Diagrams of the internal forces distribution in the pressure vessel walls can be seen in Figure 2 by these values. Due to continuous changing of the circulating curvature radius [pic], meridian force does not bear discontinuity while transferring from ellipsoid to cylinder. However, wall bending is not possible to avoid since circular force interruption from the value [pic] (head) to [pic] (cylinder) occurs on the above mentioned spot.

Radial displacements of mid-plane of the vessel wall are determined by [3, 4]:

[pic], (9)

where E is modulus of elasticity of the pressure vessel material, [pic]is relative deformation in circular direction and [pic] is Poisson’s ratio.

Substitution of (7) and (8) into (9) provides the value of the following radial head wall displacement on the joining place with cylindrical part of the vessel (with angle [pic]):

[pic], (10)

By this analogy, substitution of the values [pic] and [pic] in the formula (9) provides value of radial displacement of cylindrical vessel part:

[pic], (11)

By the aforesaid, differences in radial displacements of cylindrical part of the vessel and head occur at membrane load condition:

[pic], (12)

Since radial displacements must be continuous, reservoir wall bending occurs. From the diagram in Figure 2, it can be seen that circular forces on ellipsoidal head become negative in the area joint with cylindrical part. This can, in case of the vessel limit pressure exceeding, lead to head wall corrugation.

[pic]

Fig. 2. Membrane forces distribution at cylindrical pressure vessel with ellipsoidal heads

|[pic]Cylindrical reservoir with ellipsoidal heads, [pic] |

|[pic] |[pic] |[pic] |[pic] |[pic](MPa) |[pic](MPa) |

|0 |0 |1 |1 |138,7 |138,7 |

|* |3 |0,9959 |0,9877 |138,1 |137,0 |

|[pic] |5,625 |0,9859 |0,9575 |136,7 |132,8 |

|* |6 |0,9840 |0,9517 |136,4 |132,0 |

|* |9 |0,9650 |0,8943 |133,8 |124,0 |

|* |11 |0,9495 |0,8458 |131,7 |117,3 |

|[pic] |11,25 |0,9474 |0,8392 |131,4 |116,4 |

|* |14 |0,9223 |0,7604 |127,9 |105,4 |

|[pic] |16,875 |0,8934 |0,6676 |123,9 |92,6 |

|* |17,2 |0,890 |0,6566 |123,4 |91,0 |

|* |20,5 |0,8550 |0,5404 |118,6 |74,9 |

|[pic] |22,5 |0,8335 |0,4673 |115,6 |64,8 |

|* |24,5 |0,8122 |0,3932 |112,6 |54,5 |

|[pic] |28,125 |0,7746 |0,2582 |107,4 |35,8 |

|* |30 |0,756 |0,190 |104,8 |25,3 |

|[pic] |33,75 |0,7210 |0,0533 |99,98 |7,4 |

|* |36 |0,701 |0,0256 |97,21 |3,5 |

|[pic] |39,375 |0,6731 |-0,1396 |93,34 |-19,4 |

|* |44 |0,6392 |-0,2860 |88,63 |-39,7 |

|[pic] |45 |0,6325 |-0,3162 |87,71 |-43,8 |

|[pic] |50,625 |0,5984 |-0,4743 |82,98 |-65,8 |

|* |55,5 |0,574 |-0,5933 |79,59 |-82,3 |

|[pic] |56,25 |0,5704 |-0,6126 |79,1 |-84,9 |

|[pic] |61,875 |0,5477 |-0,7303 |75,9 |-101,3 |

|[pic] |67,5 |0,5300 |-0,8271 |73,5 |-114,7 |

|* |69 |0,526 |-0,8493 |72,9 |-117,8 |

|[pic] |73,125 |0,5166 |-0,9026 |71,6 |-125,2 |

|* |77 |0,5098 |-0,942 |70,7 |-130,6 |

|[pic] |78,75 |0,5073 |-0,9567 |70,3 |-132,7 |

|[pic] |84,375 |0,5018 |-0,9892 |69,6 |-137,2 |

|[pic] |90 |0,5000 |-1 |69,3 |-138,7 |

Table 1. Internal membrane force values in the ellipsoidal heads

3. Numerical verification

Finite element analysis of the cylindrical vessel with semi-elliptical heads has been done by using of ANSYS 10 code [5] to confirm analytical solutions. It was modeled only ellipsoidal head as axi-symmetric problem to avoid bending effects on the contact between heads and cylinder. One characteristic geometry has been chosen (R=2600 mm, h=30mm, R/H=2) subjected to the internal pressure of 1,6 MPa. Finite element mesh consists of 1200 isoparametric quadrilateral eight-node elements and 4013 nodes. Linear-elastic isotropic material law was used in the numerical calculation (E = 200 GPa, [pic]=0,3). Taking the axial symmetry into account only one half of the ellipsoid has been modeled. Tensile force with amount of [pic]has been assigned as the pressure on the contact line between cylinder and semi-elliptical head. The results for principal stresses are given in the Fig. 3.

|[pic] |[pic] |

Fig. 3. Principal stresses by cylindrical pressure vessel with ellipsoidal heads

4. Conclusions

In this paper analytical expressions for internal forces and displacements of the thin-walled cylindrical pressure vessel with semi-elliptical heads have been presented by using of membrane shell theory. It is necessary to determine these values by setting of boundary conditions equations during analysis of pressure vessels with general shell theory. The significance of the membrane components values is not only in the fact that they need for the analysis of loading capacity of pressure vessel, but also because the real stresses by regularly shaped vessel design are not much different than those obtained by membrane shell theory. Some differences appear only at the places of discontinuity of membrane displacements. Dimensionless values of internal membrane forces for geometrical parameter [pic]are given in the forms of diagram and table. Principal stresses calculated analytically are very close to the finite element results (the difference is less than 3%).

References

1] Lipschutz, M.M., Differential Geometry, Mc Graw Hill Book Company, New York, 1969.

2] Alfirević, I., Linearna analiza konstrukcija, FSB Zagreb, 2003.

3] Бидерман, В.Л., Meханика тонкостенных конструкций, Maшиностроение, Moсква, 1977.

4] Koвaленко, A.,Д., Основы термоупругости, Наукова думка, Киев, 1970.

5] ANSYS 10.0, User’s manual, 2005.[pic]

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