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Estimation of stresses at knuckle of variable thickness in torispherical heads.


Pressure vessels are finding applications in several fields such as mechanical, chemical, aerospace engineering. They are used to store liquids at high pressure in technological applications. In general, they are of circular cylindrical shape with end domes (heads). Stress perturbations occur at the junction between the cylinder and the dome due to change of geometry. Such perturbation is minimum, if hemispherical geometry is used for dome. In certain advanced technological applications (rockets used in space craft launches), in order to limit the length of the vehicle, torispherical heads are employed. Two examples of current use are Indian Space Research Organization's (ISRO) Geosynchronous Satellite Launch Vehicle (GSLV) and Space shuttle program of US Space Agency, NASA. The Solid Rocket Boosters (SRBs), liquid and cryogenic stages use the torispherical heads in the tankages.

This paper presents an analysis of stresses in the knuckle region of pressure vessel with torispherical head. Due to change of geometry, high stresses are induced in this region and net compressive stresses is in the hoop direction and severe bending stress is in the axial direction. The causes of worry to designers are (i) buckling due to compressive stress (ii) the material failure to high bending stress regions. The torispherical geometry is composed of two circular arcs; (i) the torus region (knuckle) of smaller radius with center of curvature located off the centroidal axis and (ii) the spherical dome (crown) with larger radius as shown in Fig. 1. The ratio of the above two radii controls the nature of stress field in the knuckle region and hence is of importance to the designers. Usually, the thickness of knuckle region is taken to be equal to that of crown and cylinder. This gives rise to very high stress level in the knuckle region. In this paper, an attempt is made to minimize the stress level by providing variable thickness in the knuckle region. As it is difficult to obtain classical shell theory based solution, the numerical method namely, finite element solution is employed.


In the view of the technological importance of pressure vessel with torispherical head, several studies have been carried out and reported in the literature. There are several codes of practice governing the design of pressure vessel. namely, IS 28251969, ASME Pressure vessel code 2004 section VIII and BS 1515. It was Galletly [1] who pointed out the deficiencies in the design as per the codal provisions when torispherical heads are used. According to him, the code predicted the stresses in the knuckle, which were less than one half of those actually occurring and gave no indication of possibility of failure by yielding and also buckling due to circumferential compressive stress. Fessler and Stanley [2-3] carried out experimental studies on torispherical pressure vessel heads using photo elastic model. They pointed out the dependence of the elastic stress distribution on the shape and thickness parameters in a wide range of torispherical heads. The wall thicknesses of specimens were equal to that off cylinder. They designed 32 nos. of test models having the wall thickness to cylinder radius ratio ([R.sub.c] / h) in the range of 4.5 to 33.33 and the head height to cylinder radius ratio ([R.sub.c] / H) were varied from 1.142 to 5. The peak principle stress indices were presented in the form of two contour systems and in terms of mean and bending stresses. They computed the maximum bending stresses in the knuckle region in all models and presented an empirical equation relating peak elastic stress, the head height and the wall thickness ratios in a range of torispherical heads. Ganapathy and Charles [4] analyzed the stresses in torispherical pressure vessel by classical asymptotic expansion. They arrived at simple design formula to find the maximum stress in the toroidal segment, instability pressure and optimum design of the toroidal knuckle. They compared the numerical results with experimental results available in the literature and have reported a close agreement between the two. Stanley and Campbell [5-6] carried out experimental studies on torispherical head pressure vessels subjected to internal pressure. They prepared the model in an austenitic stainless steel material with nominal diameters equal to 54, 81, and 108 inches and thickness to radius ratios (h / [R.sub.c]) equal to 21.1, 316 and 420. They predicted that the greatest stress index (stress index I is the ratio of the principle stress at a point to nominal hoop stress in the cylinder) in each case was that of the hoop compression in the knuckle. The inner surface meridional stress index reached the peak in the knuckle and was tensile in nature. The hoop stress index on the inner surface was greater than that of the outer surface in the knuckle. They also concluded that no simple correlation was apparent between the peak stress indices and the thickness ratios for the different crown shapes. For an increase in knuckle radius to cylinder radius ratio ([R.sub.d]/[R.sub.c] and [h.sub.d]/[R.sub.c] remaining constant) the peak stress indices were found to decrease. It may be observed that as the ratio of knuckle-to-cylinder radius increases, the geometry of the crown tends to behave as that of hemispherical dome. The variations [I.sup.i.sub.[phi]] where I = inner surface and [phi] = Axial / meridional coordinate) and [I.sup.o.sub.c] where o = outer surface and c = circumferential coordinate) with [R.sub.d] / [R.sub.c] ([R.sub.k] / [R.sub.c] and [h.sub.d] / [R.sub.c] remaining constant) followed no simple consistent trend.

Sori'c [7] investigated initial imperfection sensitivity of the torispherical shells subjected to an internal pressure by finite element method. He used rectangular doubly curved finite element with 48 degree of freedom per element using the software, FEMAS. Torispherical shells failed to demonstrate any significant imperfection sensitivity for the imperfection pattern analysed. The imperfection shape was assumed to be that of first buckling mode. The magnitude of imperfection was taken to be 1/10th of wall thickness. The load carrying capacity of imperfect shell was found to be about 8% less than that of the perfect geometry. For the imperfection amplitude equal to one half of the wall thickness, the reduction in the load was found to be about 15%. Blachute [8] carried out experimental investigation on torispherical head models made of mild steel with five sets of geometry parameters keeping constant cylinder diameter (= 200mm) and constant wall thickness (= 6mm). The models were subjected to an internal pressure and deflection of the apex point was measured. From the plots of apex deflection vs. internal pressure, yield pressure was obtained and was compared with analytical values. He did not investigate the nature of stresses at the knuckle region where the yield was expected to occur. Theoretically calculated plastic collapse pressure differed by not more than 12% from those obtained by experiments. Tafreshi and Thorpe [9-10] carried out investigations on the design sensitivity of geometric imperfection of thin torispherical headed vessel, subjected to an internal pressure by finite element method. They used the experimental data available in literature [5, 6] as input. Three node axisymmetric shell element, designated as SAX2 in the ABAQUS (a general purpose finite element) software, was used for modeling. The deformed profiles showed inward displacement in the knuckle part. For the analysis, the mean dimensions of the dome and cylinder were considered throughout. Using the experimental data [5, 6], they analyzed the stresses in perfect and imperfect models. They concluded that there was a distinction between the real and nominal dimensions (imperfect and perfect geometries) of pressure vessel. Finite element models of end domes considering curvature and thickness imperfection were analyzed and stresses were compared with the results of perfect domes and with the corresponding experimental results. With imperfection, the stress indices were shown to differ by significant amounts (up to 60%) from those to be expected in perfectly formed vessel. The greatest changes were due to curvature variations but the effects of thickness variation are also significant.

The above review of literature reveals that: (i) several investigations were concerned with pressure vessel of torispherical geometry uniform thickness in the knuckle region mostly equal to that of cylinder thickness, (ii) the radius to thickness ratios considered were of 4.5, 21.1, 33.33, 316 and 420. First two thickness ratios of shells may be characterised as a thick shells, normally used in mechanical engineering applications and (iii) stress indices were found to be high indicating failure in the knuckle regions. Present work is concerned with studies on thin walled pressure vessel, which are used in aerospace applications. The knuckle region is of variable thickness in order to reduce the stress levels. As exact solutions are not available, finite element analysis has been employed. Results are presented for various combinations of ratios of crown to knuckle radii and that of knuckle to cylinder wall thickness. The results are compared with that of Fessler [2-3]on qualitative basis.

Description of geometry Problem definition

The torispherical head geometry is composed of two circular arcs, crown of radius [R.sub.d] at the top and knuckle of radius [R.sub.k]. This is fitted on the cylinder of radius [R.sub.c]. With reference to Fig. 1, [O.sub.d] is the center of curvature of the crown and [O.sub.k] is the center of curvature of the knuckle. The angle subtended by the knuckle at [O.sub.k] is computed as:

[alpha] = Cos - [[R.sub.c]-[R.sub.k] / [R.sub.d]-[R.sub.k]] (1)

This ensures the continuity of slopes at 'a', the junction of crown and knuckle. The rise of head, H is computed by:

H = [R.sub.d] - [square root of [(([([R.sub.d]-[R.sub.k]).sup.2] - [([R.sub.c]-[R.sub.k]).sup.2]))]. (2)

[h.sub.c], [h.sub.d], and [h.sub.k] are the thicknesses of cylinder, crown and knuckle respectively. For the case of uniform wall thickness, we have [h.sub.c] = [h.sub.d] = [h.sub.k]. In the description of the above geometry, the following two parameters are defined, (i) the ratio of crown-to-knuckle radii, [beta] = [R.sub.d]/[R.sub.k] and (ii) the ratio of knuckle-to-cylinder thicknesses, [gamma] = [h.sub.k]/[h.sub.c]. For the case of pressure vessel with uniform wall thickness, [gamma] = 1. In the case of hemispherical dome, [beta] = 1.

Knuckle of Variable Thickness

The inner boundary of the knuckle is a smooth curve. There is no discontinuity in the geometry at both junctions; cylinder-knuckle and crown-knuckle. The radius of curvature of inner surface [R.sub.k] (knuckle radius) is constant and is equal to [R.sub.d]/[beta], where [R.sub.d] is dome radius, (Fig. 1.). However the meridional curvature of outer surface is of variable geometry, which is obtained by Spline curve method

Spline Curve Method

With reference to Fig. 2 are the junctions of outer surface of crown-knuckle and cylinder-knuckle respectively. The bisector of the angle [alpha] subtended by the knuckle at its center of curvature is drawn. J is the point of intersection of the bisector with the inner surface. Fix the point D on the bisector at a distance equal to [gamma]h from J. Using spline curve method; join A, D and B. ADB is the outer surface of the knuckle. It may be noted that there are discontinuities of slope in the cylinder-knuckle and crown-knuckle junctions. The maximum thickness of the knuckle J-D is equal to [gamma]h.


Axisymmetric analysis

If the geometry, loading and boundary conditions possess axial symmetry, the problem is defined as axisymmetric. All pressure vessels satisfy these conditions, if cut-outs and opening are ignored. Even though a considerable simplification results in the solutions of axisymmetric problems, exact solution is possible only in the simple case of circular cylindrical shell with hemispherical dome. Analytical expressions are available for the stresses in the junction region between the cylinder and hemispherical dome [15]. In case of torispherical head, due to the complexity of deformation (cylindrical and dome deforming in the opposite directions), it is not possible to obtain analytical solution. Only approximate solutions are possible. The finite element method, which is a numerical procedure, is suitable for solving many complex structural problems. Finite element modeling of pressure vessel can be in three ways: using (i) general shell element, (ii) axisymmetric shell element and (iii) axisymmetric solid element. From numerical accuracy point of view, the later two are suitable. The axisymmetric solid element is suitable for analysing the structural problems, both thick and thin shells. For many practical applications, the geometrical data are such that the radius-to-thickness ratios of cylinder and torispherical head are greater than 200. These two parts are considered as thin shells. When applying variable thickness in knuckle portion is to reduce stress level, the radius-to-thickness ratio reduces considerably, about to 20. According to literature [15-16]; radius-to-thickness ratio greater than 30 is termed as thin shells. The geometric parameters in the knuckle region particularly with variable thickness do not satisfy this condition. Hence it is appropriate to use axisymmetric solid FE modeling to analyse the stress field. As seen from the literature, the axisymmetric solid FE modeling has been carried out in four ways; using (i) triangular element (three nodes at the apex termed as constant strain triangle (CST) and six node including three midside nodes, termed as higher order triangular element), (ii) four-node linear quadrilateral element, (iii) four-node quadrilateral element with incompatible mode and (iv) eight-node quadratic element with four midside nodes. The triangular elements are not suitable for varying stress field conditions (severe bending state of stress in the knuckle region). In the present work, the latter three elements are considered in the estimation of stresses in the critical region of torispherical head. In the following, we briefly describe the above three elements [17].

Four node linear quadrilateral element

With reference to Fig.3, the element geometry is described by means of four nodes in terms of global coordinates ([x.sub.i], [y.sub.i]). Each node has two degrees of freedom (d.o.f) [u.sub.i] and [v.sub.i], taking the total d.o.f per element to eight. The displacement component (u, v) and the geometry at any point within the element are described, using shape function, [N.sub.i] as follows.

{u v} = [4.summation over (i=1)] [N.sub.i] ([epsilon],[eta]) {[u.sub.i] [v.sub.i]} (3)

{x y} = [4.summation over (i=1)] [N.sub.i] ([epsilon],[eta]) {[x.sub.i] [y.sub.i]} (4)

where the shape functions

[N.sub.1] = 1/4 (1-[xi])(1-[eta]); [N.sub.3] = 1/4 (1 + [xi])(1+[eta])

[N.sub.2] = 1/4 (1+[xi])(1-[eta]); [N.sub.4] = 1/4 (1-[eta])(1+[eta])


Four-node linear quadrilateral element with incompatible mode

With reference to Fig. 3, the element with incompatible mode, the coordinates of a point within the element are described in the same way as given in Eq. (4). However, the displacement field u and v described as follows.

u = [4.summation over (i=1)][N.sub.i][u.sub.i] + (1-[[xi].sup.2])[a.sub.1] + (1-[[eta].sup.2])[a.sub.2]

v = [4.summation over (i=1)][N.sub.i][v.sub.i] + (1-[[xi].sup.2])[a.sub.3] + (1-[[eta].sup.2])[a.sub.4] (5)

The [N.sub.i] are the same shape function of the four-node element, and the four [a.sub.i] are the generalized d.o.f., not associated with any node, nor are they connected to d.o.f of any other element. Physically, displacement modes associated with the [a.sub.i] are displacement relative to the displacement field dictated by the summation in Eq. (5). i.e it is a quadrilateral element with six shape functions.

Eight node quadratic element

With reference to Fig. 4, the element geometry is described by means of eight nodes (four corner nodes and four midside nodes) in terms of global coordinates ([x.sub.i], [y.sub.i]). Each node has two degrees of freedom [u.sub.i], and [v.sub.i], taking the total degree of freedom (d.o.f) per element is to sixteen. The displacement components (u, v) and the geometry at any point within the element are described, using shape function, [N.sub.i] as follows.

{u v} = [8.summation over (i=1)] [N.sub.i]([epsilon],[eta]) {[u.sub.i] [v.sub.i]} (6)

{x y} = [8.summation over (i=1)] [N.sub.i]([epsilon],[eta]) {[x.sub.i] [y.sub.i]} (7)

where the shape function [N.sub.i]

[N.sub.1] = - (1-[xi])(1-[eta]) (1+[xi]+[eta]) / 4;

[N.sub.2] = - (1+[xi])(1-[eta]) (1-[xi]+[eta]) / 4

[N.sub.3] = - (1+[xi])(1+[eta]) (1-[xi]-[eta]) / 4;

[N.sub.4] = - (1-[xi])(1+[eta]) (1+[xi]-[eta]) / 4

[N.sub.5] = (1-[[xi].sup.2])(1-[eta]) / 2;

[N.sub.6] = (1+[[xi])(1-[[eta].sup.2]) / 2

[N.sub.7] = (1-[[xi].sup.2])(1+[eta]) / 2;

[N.sub.8] = (1-[[xi])(1-[[eta].sup.2]) / 2

The equilibrium equations, at the element level are obtained using the principle of minimization of total potential. The size of the element stiffness matrix for above three elements varies. For four-noded quadrilateral element, it is 8 x 8 matrix. In the case of quadrilateral element with incompatible mode, it is initially 12 x 12 matrix. The four-extra d.o.f. [a.sub.1] to [a.sub.4] (nodeless variables) are condensed out, and the resulting stiffness matrix reduces to 8 x 8 size. In the case of eight-node quadratic element, the stiffness matrix is of 16 x 16 size. The introduction of incompatible mode, (the shape functions associated with [a.sub.1] to [a.sub.4]) violates the inter element displacement compatibility. In spite of, this element has been found to perform well in the varying stress field condition, as they pass the patch test [17].


The pressure vessels considered is of circular cylindrical geometry with torispherical dome. In order to reduce the stress level, the knuckle region is of variable thickness can be obtained in spline curve method and tangent method. The previous study carried out by the present authors [14], indicated that the geometry of variable thickness obtained by spline method was more effective, giving rise to lower von-Mises stress level. Hence the present study, we considered the knuckle of variable thickness obtained by spline method [2.3].

The FEM, being a numerical process, the numerical accuracy obtained depends on the meshing. Three types of meshing [11] are considered for the present study: (i) single element across the thickness, Fig. 5a. (model 1), (ii) two element across the thickness, Fig. 5b (model 2), and (iii) four element across the thickness, Fig. 5c (model 3). The numerical investigations are carried out for the stress field in the knuckle region under nine cases (three elements multiplied by three meshes), selected cases of geometric parameters, ratio of crown-to-knuckle radius ([beta]) and ratio of knuckle-to-cylinder thickness ([gamma]).




Numerical studies--Data

The geometric parameters of a characteristic solid rocket booster (SRB) are used. The SRB is composed of circular cylindrical shell with torispherical domes and is made of maraging steel. Geometrical parameters are: [R.sub.c] = 1600 mm, [R.sub.d] = 1750 mm, [h.sub.c] = [h.sub.d] = 8 mm (Figure 1). Material properties are: Young modulus, E = 180 GPa, Yield stress, [[sigma].sub.y] = 1482 MPa, Poisson's ratio, v = 0.3, The SRB is analysed for an internal pressure (p) of 6.5 MPa ([approximately equal to] 1000 psi). Table 1 and 2 give the value of the two parameters ([beta],[gamma]) and used in the numerical examples.

[alpha] and H are computed using the Eqs. (1) and (2).The axisymmetric finite element model of the pressure vessel with torispherical head is shown in Fig.6. At the bottom edge, the axial displacement is constrained. At the apex point radial displacement is constrained. For the simplest model, four-nodded quadrilateral element with one element across the thickness, The FE mesh consists of 784 nodes and 391 elements. On the other extreme, eight-nodded quadratic element with four elements across the thickness, the FE mesh consists of 5497 nodes and 1568 elements. An internal pressure(p) of 6.5 Mpa is applied.


Result and discussion

Deformed Shapes

Deformed shape of the pressure vessel with internal pressure is shown in Figure 7., for [beta] = 3.36, [gamma] = 1 (uniform thickness). It may be observed that the knuckle region moves inward. This gives rise to the severe bending state of stress across the thickness with compressive stress in the outer surface and tensile stress in the inner surface in the meridional / axial direction. Add it been a hemispherical instead of torispherical head, it has been shown the knuckle region moves outward, but by different amount [15].


Stresses in Knuckle

Stress analyses have been carried out on pressure vessels with torispherical and hemispherical heads. The stress levels in the junction region are important to the designer. In order to reduce the stress levels at this critical zone, knuckle of variable thickness has been suggested. The inner surface is of circular arc, of radius [R.sub.k] and the outer surface is obtained by spline method. The meridional/axial stress ([sigma][phi]) assumes important. Stresses have been obtained under forty five cases: three axisymmetric solid elements with three meshes (Model 1, Model 2 and Model 3, Fig. 7.) and for give values of knuckle to cylinder thickness ratio ([gamma]). The nature of stress field in the junction for uniform thickness ([gamma]=1) in knuckle is shown in figs. 8a and b. The stresses obtained using eight node element normalized with respect to hoop stress ([[sigma].sub.c]) have been plotted in figures (along the meridional coordinate from the pole(s)). The stress variation exhibits symmetric about the centre line of knuckle and also the stress bulbs are symmetric about the nominal axial stress. The outer surfaces experiences the compressive stress and inner surface experiences the severe tensile stress which is about 2.2 times the nominal axial stress (= p [R.sub.c] / 2h). In other words, the axial stress is greater than the nominal hoop stress ([[sigma].sub.n] = p [R.sub.c] / h).. The fig. 8a is Model 1 (1 element across thickness) and fig. 8b is for Model 3 (4 element across the thickness). Figs. 8c and d are stress plot for the case of knuckle of variable thickness [gamma]=2.5. Using eight node element for the cases Model 1 and Model 3 respectively, it may be seen that the use of variable thickness help to reduce the stress levels in the critical region, there is a sudden dip





Von Mises Stresses

In the design of structures, two yield criteria namely von Mises and Tresca are in use. The von Mises criterion is based on the distortion energy theory and the Tresca is based on the maximum shear stress theory. Among the two, the von Mises is seen to be more popular. The von Mises stress index, [I.sub.von] is defined as ([square root of ([sigma].sup.2.sub.[phi] + [[sigma].sup.2.sub.c] - [[sigma].sub.[phi]] * [[sigma].sub.c])]) / [[sigma].sub.y] where [[sigma].sub.y] is the yield stress of the material under uniaxial tension. It is less than one for failure not happen an equal to one for yielding starts. Keeping in this mind, the peak values of [I.sub.von] is obtained for the 45 cases, analysed in this work are given Table 3. In addition, the values of [I.sub.von] obtained axisymmetric shell element are also presented in this table 14. The geometric parameters assumed are [beta] = 3.36 (Torispherical head) and [gamma] varied from 1 to 3. The table gives the value of [I.sub.von] for four types of element and for 3 types of meshes. As seen from the table for [gamma] = 1, the yielding occurs in the inner surface, as [I.sub.von] is greater than 1 and yielding does not occur in outer surface. This may be explained as follows. As seen from the figs. 8a and b, the inner surface is tensile nature. However, the hoop stress is of compressive nature (as presented in the earlier study) [14]. This condition of opposite nature of stress at a point makes the yield criteria satisfied much earlier. On the other hand the axial stress and hoop stress in the outer region are both compressive nature. The same sign of stresses delay the satisfaction of yield criterion and hence the [I.sub.von]s for the outer surface are less than unity.

When knuckle of variable thickness used [gamma] > 1, it is seen that [I.sub.von] for both inner and outer surfaces are lower than 1. That means yield does not occur when the knuckle of various thickness is employed. Comparison of [I.sub.von] for particular mesh (example Mesh.3). The values obtained by four node in compatible mode and eight node elements are more or less equal are higher than that obtained by four node element. Hence, it is observed that the performances of four node element with incompatible mode and the quadratic eight node element are comparable. From the point of view of vide difference in the size of the problem (number of equations to be solved and larger band width) use of four node quadrilateral element with incompatible node is always advantageous from both numerical accuracy and computational efficiency. Also it may be noted that, the values of [I.sub.von] credited by the shell element are always greater than those obtained by three axisymmetric solid elements.


The following observation are made in the previous section

(1) From the point of view of vide difference in the size of the problem (number of equations to be solved and larger band width) use of four node quadrilateral element with incompatible node is always advantageous from both numerical accuracy and computational efficiency. Also it may be noted that, the values of [I.sub.von] credited by the shell element are always greater than those obtained by three axisymmetric solid elements.

(2) Using eight node element for the cases Model 1 and Model 3 respectively, it may be seen that the use of variable thickness help to reduce the stress levels in the critical region.

(3) Higher values of [beta] is desirable from the point view of reducing the vessel length, particularly in certain critical applications such as rocket motor casing; shorter the length better the control of vehicle during launch.

(4) In the torispherical domes [beta] > 1, the net compressive hoop stress is experienced in the knuckle region which is of concerned to the designer as it will lead to failure by buckling. Use of knuckle of variable thickness is expected to avoid this condition; however this issue has not been addressed in this paper.

(5) As the dome radius to knuckle radius ([beta]) becomes larger than the one, the dome rise (H) decreases.

(6) Inward displacement in knuckle region is considerably reduced when applying variable thickness in knuckle


[1] Galletly, G.D., 1959, "Torispherical shells--A caution to designers," J Engineering for industry Trans ASME; 81, pp. 51-62.

[2] Fessler, H., and Stanley, P., 1965, "Stresses in torispherical drum heads: a photo elastic investigation," J Strain Analysis, 1(1), pp. 69-82.

[3] Fessler,H., and Stanley P., 1966, "Stresses in torispherical drum heads: a critical evaluation," J Strain Analysis, 1(2), pp. 89-101.

[4] Ganapathy, V.R., and Charles, R.S., 1985, "Analysis of torispherical pressure vessels," J Engineering Mechanics, Trans ASCE, 102(4), pp. 643-657.

[5] Stanley, P., Campbell, T.D., 1981 "Very thin torispherical pressure vessel ends under internal pressure: test procedure and typical results," J Strain Analysis, 16(3), pp. 171-186.

[6] Stanley, P., Campbell, T.D., 1981, "Stanley P, Campbell TD. Very thin torispherical pressure vessel ends under internal pressure: strains, deformations and buckling behaviour," J Strain Analysis, 16(3), pp. 187-203.

[7] Sori'c, J., 1995, "Imperfection sensitivity of internally pressurized torispherical shells," Thin Walled Structures, 23, pp. 57-66.

[8] Blachute, J., 1995, "Plastic loads for internally pressurized torispheres," Int J Pressure Vessels & piping, 64 pp. 91-100.

[9] Tafreshi, A., and Thorpe, T.E., 1996, "Effects of local departures from nominal dimensions on stresses in thin torispherical end closures" J Strain Analysis, 31(4), pp. 315-324.

[10] Tafreshi, A., 1997, "Numerical analysis of thin torispherical end closures," Int J Pressure Vessels & piping, 71, pp. 77-88.

[11] Ruicheng Tong, and Xucheng Wang, 1997, "Simplified method based on the deformation theory for structural limit analysis-II. Numerical application and investigation on mesh density," Int J Pressure Vessels & piping, 70, pp. 51-58

[12] Mahendra D.Rana, and Arturs Kalnins, 2000, "Technical basis for code cases on design of ellipsoidal and torispherical heads for ASME section VIII vessels," J Pressure Vessel Technology TransASME, 122, pp. 55-59.

[13] Yeom, D., and Robinson, M., 1996, "Numerical analysis of the elastic-plastic behaviour of pressure vessels with ellipsoidal and torispherical heads," Int J Pressure Vessels & piping, 65, pp.147-156

[14] Kanagasabapathy, H., and Chockalingam, Kn.K.S.K., "Numerical Analysis of Stresses at Knuckle of Variable Thickness of Torispherical Heads," Submitted for publication

[15] Timoshenko, S., and Krieger, S.W., 1959, Theory of plates and shells, 1st ed. New York, McGraw-Hill International Book Company.

[16] Harry Kraus, 1967, Thin Elastic Shells, New York, John Wiley & Sons, Inc.

[17] Cook, R.D., Malkus, D.S., Plesha, M.E., and Witt, R.J., Concept and applications of finite element analysis, 4th ed, New York, John Wiley & Sons, Inc.

[18] Tirupathi R Chandrupatla, and Ashok D Belegundu, 2004, Introduction to finite elements in engineering, 3rd ed. New Delhi, Prentice-Hall of India Private Limited.

H. Kanagasabapathy * and K. S. K. Chockalingam

Department of Mechanical Engineering, National Engineering College, K. R. Nagar, Kovilpatti, Tamil Nadu, India. Pin: 628 503

* Corresponding author Email address:
Table 1: Nominal geometric parameters (see Fig. 1.).

                              Subtended       Head
Sl.        [beta] =         angle [alpha]    rise H
No    [R.sub.d]/[R.sub.k]     (degree)         mm

1             1.5               42.02         1359
2              2                34.05         126
3              3                29.37         1178
4            3.36               28.6          1162
5              4                27.66         1141
6              5                26.77         1120

7             10                25.21         1079

Table 2: Thickness ratio of knuckle to cylinder.

                [gamma] =
Sl.No      [h.sub.k]/[h.sub.c]

1                   1

2                  1.5

3                   2

4                  2.5

5                   3

Table 3: von Mises stress criteria in knuckle region obtained by
variable thickness-spline curve method.

         [gamma] =                         [I.sub.von] =
[beta]   [h.sub.k]/                     ([[sigma].sub.von]/
         [h.sub.c]                        [[sigma].sub.y])
                      Element type       MESH - I (Fig. 5a)

                                      Outer            Inner

                      4node           0.8008           1.057
                      4node(incomp)   0.8596           1.1568
3.36         1
                      8node           0.86             1.1579
                      3 node          0.8635           1.1538
                      4node           0.7961           0.8382
                      4node(incomp)   0.8445           0.9411
3.36        1.5
                      8node           0.8446           0.9428
                      3 node          0.8494           0.9452
                      4node           0.7927           0.7988
                      4node(incomp)   0.8343           0.8463
3.36         2
                      8node           0.8343           0.8498
                      3 node          0.8401           0.8702
                      4node           0.7904           0.7962
                      4node(incomp)   0.8276           0.822
3.36        2.5
                      8node           0.8275           0.8302
                      3 node          0.8338           0.8785
                      4node           0.7889           0.7946
                      4node(incomp)   0.8233           0.8224
3.36         3
                      8node           0.8232           0.8409
                      3 node          0.8297           0.9191

             [I.sub.von] =                   [I.sub.von] =
[beta]     ([[sigma].sub.von]/             ([[sigma].sub.von]/
            [[sigma].sub.y])                [[sigma].sub.y])
           MESH - II (Fig. 5b)             MESH - III (Fig. 5c)

         Outer           Inner           Outer           Inner

         0.8145          1.0906          0.8257          1.1043
         0.8596          1.1591          0.8596          1.1597
         0.8598          1.1598          0.8598          1.1603
         0.8635          1.1538          0.8635          1.1538
         0.8081          0.883           0.8176          0.9013
         0.8444          0.9449          0.8445          0.9461
         0.8445          0.9459          0.8445          0.9467
         0.8494          0.9452          0.8494          0.9452
         0.8034          0.7933          0.8118          0.7967
         0.8343          0.8509          0.8342          0.8526
         0.8342          0.8538          0.8342          0.8549
         0.8401          0.8702          0.8401          0.8702
         0.8002          0.7911          0.8077          0.7894
         0.8275          0.8272          0.8274          0.8297
         0.8274          0.8328          0.8274          0.8343
         0.8338          0.8785          0.8338          0.8785
         0.798           0.7898          0.805           0.7882
         0.8232          0.8283          0.8232          0.831
         0.8232          0.839           0.8231          0.8405
         0.8297          0.9191          0.8297          0.9191

Note: 4node--Four-node quadrilateral axisymmetric solid element,
4node(incomp)--Four-node quadrilateral axisymmetric solid element with
incompatible mode option, 8node--eight-node quadratic axisymmetric
solid element, 3 node--Three-node quadratic axisymmetric shell element
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Author:Kanagasabapathy, H.; Chockalingam, K.S.K.
Publication:International Journal of Applied Engineering Research
Article Type:Report
Date:May 1, 2009
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