Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 48, 2, pp. 297-307, Warsaw 2010 SHAPING OF A MIDDLE SURFACE OF A DISHED HEAD OF A CIRCULAR CYLINDRICAL PRESSURE VESSEL Jerzy Lewiński Poznan University of Technology, Institute of Applied Mechanics, Poznan, Poland e-mail: jerzy.lewinski@put.poznan.pl Krzysztof Magnucki Poznan University of Technology, Institute of Applied Mechanics, Poznan, Poland Institute of Rail Vehicles, Tabor, Poznan, Poland krzysztof.magnucki@put.poznan.pl The paper is devoted to a dished head of a pressure vessel subject to in- ternal uniformpressure.A short survey of optimal design of the pressure vessel and its head is presented. The problem of shaping of the middle surface of the dished head with the use of trigonometric series is depic- ted. As a criterion of the shaping process, the continuity of curvatures of the surfaces in the joint of the circular cylindrical shell and the dished head is assumed.Results of the numerical calculation for optimal shapes of the head are presented in figures. Key words: thin-walled pressure vessel, dished head,minimal stress con- centration 1. Introduction The standard torispherical, ellipsoidal or hemispherical head of a pressure ves- sel significantly disturbs themembrane stress pattern arising in its cylindrical part. The value of the meridional principal curvature of the middle head sur- face is non-zero while in the cylindrical it takes the zero level. In result, the curvature becomes discontinuous. The problem of dished heads of the vessels has been undertaken by many investigators. Middleton (1979) presented an optimal design problem of the torispherical pressure vessel head with the use of the penalty function procedure. Mansfield (1981) proposed the meridian shape in the form of an integral equation determining the optimal surface of revolution and compared the results with classical ellipsoidal and torispheri- cal heads. Yushan and Wang (1996), Yushan et al. (1996) calculated stresses 298 J. Lewiński, K. Magnucki of ellipsoidal heads and noticed the stress concentration occurring there. Ma- gnucki and Lewinski (2000) described the stress state arising in an untypical torispherical head composed of circular and polynomial parts.Magnucki et al. (2002) solved the problem of stress minimization of a vessel with an ellipso- idal head. Magnucki and Lewinski (2003) presented the optimal design of an ellipsoidal headwith consideration of various thickness values of the shell.Ma- linowski andMagnucki (2005)minimized the stress concentration in sandwich ribbed flat baffle plates of a cylindrical tank. Krivoshapko (2007) presented a review of strength and buckling problems of generalized and ellipsoidal shells of pressure vessels. Liu et al. (2008) proposed a theoreticalmethod using finite element analysis to calculate theplastic collapse loads of pressurevessels under internal pressure and compared the analytical methods according to three cri- teria stated in theASMEBoiler PressureVessel Code.Błachut sandMagnucki (2008) delivered a review of strength, static stability, and structural optimiza- tion of horizontal pressure vessels. Wittenbeck and Magnucki (2008) shaped the dished headmeridian in the form of clothoidal and circular parts. Ventsel andKrauthammer (2001) delivered amonograph presenting the strength and stability problems of plates and shells with the edge effect of cylindrical shells. The present paper is a continuation of the strength and optimization pro- blems and deals with the shaping of the middle surface of dished heads with the use of trigonometric functions. 2. Mathematical description of the middle surface of the dished head Theshapeof ahead closinga cylindrical pressurevessel significantly affects the pattern of stress arising along its meridian. Since the stress depends, among others, on the meridian curvature, its radius of curvature should be continu- ous. Curvature of commonly used torispherical or ellipsoidal heads undergoes sudden variation in the contact point of the head and cylindrical parts of the meridian. This is due to the fact that the meridional curvature radius of the cylindrical part is equal to infinity, while the further course of the meridian belonging to the head is of a finite radius. In order to avoid such a situation, the head profile should begin from the infinite radius too. Such a shape of the headmay be described by the following functions r(z)= ar̃(ζ) (2.1) where Shaping of a middle surface of a dished head... 299 r̃(ζ) – dimensionless radius, r̃(ζ)= f0(ζ)=α1cos(πζ)+α2cos(2πζ)+α3cos(3πζ) ζ – dimensionless coordinate, ζ = z/b0 b0 – size (a linear quantity) a – radius of the cylindrical shell. The continuity conditions of the dimensionless radius for the joint of cy- lindrical shell and the dished head have the following form r̃(0)= 1 giving α1+α2+α3 =1 (2.2) The other conditions that should be met by function (2.1) in order to ensure stepless variation of the radius are as follows dr̃ dζ ∣∣∣∣ ζ=0 ≡ 0 fulfilled by identity (2.3) Thefirst of theabove equations is satisfiedby identity,while the otherprovides another condition for α1, α2, and α3 d2r dζ2 ∣∣∣∣ ζ=0 =0 giving α1+4α2+9α3 =0 (2.4) This allows one to express the coefficients α2 and α3 in terms of α1 α2 = 1 5 (9−8α1) α3 = 1 5 (3α1−4) (2.5) Thus, function (2.1) may smoothly match the cylindrical part of the vessel shape, but in order to provide a satisfactory shape of the head it must be completed by a circular part. Since the connection between the cosinusoidal and circular parts of the meridian should be smooth too, the circle should begin in the point at which the centre of curvature of cosinusoidal curve (2.1) reaches the axis of vessel symmetry. The longitudinal-meridional curvature radius is Rm = [ 1+ ( dr dz )2]3/2 ∣∣∣d 2r dz2 ∣∣∣ = a [ 1+ ( π β0 )2 f21(ζ) ]3/2 ( π β0 )2 f2(ζ) (2.6) and the circumferential-parallel curvature radius Re = r(z) cosθ = af0(ζ) √ 1+ ( π β0 )2 f21(ζ) (2.7) 300 J. Lewiński, K. Magnucki where f1(ζ)=α1 sin(πζ)+2α2 sin(2πζ)+3α3 sin(3πζ) f2(ζ)=α1cos(πζ)+4α2cos(2πζ)+9α3cos(3πζ) β0 = b0 a Fig. 1. Exemplary location of the centre of curvature for the point M of the cosinusoidal curve In any point of the considered curve its derivative equals the tangent of the angle θ ((dr/dz)−1 =tanθ). Hence, according to Fig.1, one might easily write the following expressions for the coordinates of the centre of curvature zc = z− 1+ ( dr dz )2 d2r dz2 dr dz = aβ0 { ζ− f1(ζ) πf2(ζ) [ 1+ ( π β0 )2 f21(ζ) ]} (2.8) rc = y+ 1+ ( dr dz )2 d2r dz2 = a { r̃(ζ)− β20 π2f2(ζ) [ 1+ ( π β0 )2 f21(ζ) ]} Taking into account relationship (2.5), cosinusoidal curve (2.1) is determined by two parameters: α1 and b0. Therefore, once their values are assumed, one is able to find such a point M at the cosinusoidal part of the curve fromwhich the circular shape begins (Fig.2). In consequence, selection of pairs of α1 and b0 parameters enables finding a family of head shapes of various values of the relative depth β given by the formula β= b a =β0ζM + R̃m(1+cosθM) (2.9) where ζM = zM b0 R̃m = Rm a Shaping of a middle surface of a dished head... 301 Fig. 2. The head shape composed of the cosinusoidal and circular parts connected at the point M 3. Equivalent stress of the circular cylindrical vessel The longitudinal and circumferential stresses of the head are as follows σm = 1 2 Re p0 t σe = 1 2 Re ( 2− Re Rm )p0 t (3.1) where p0 is the uniformly distributed pressure, t – thickness of the head. The equivalent stress (i.e. Huber-Mises stress) is σeq = 1 2 Re √ 3−3 Re Rm + (Re Rm )2p0 t or σeq = σ̃eqa p0 t (3.2) where the dimensionless equivalent stress is σ̃eq = 1 2 R̃e √ 3−3Re Rm + (Re Rm )2 and R̃e = Re a (3.3) Ventsel and Krauthammer (2001) described theory and applications of thin plates and shells, with consideration of the edge effect in thin-walled shells. An exemplary variant of the head obtained for α1 = 0.8, α2 = 0.52, α3 = −0.32 is shown in Fig.3. Distribution of the dimensionless equivalent stress of this head is shown in Fig.4. The centre of curvature runs along its trajectory and intersects twice with the x-axis. This provides two possible solutions for the head. In the case of 302 J. Lewiński, K. Magnucki Fig. 3. Example of the head solution for α1 =0.8 and β0 =2 Fig. 4. Dimensionless equivalent stress for α1 =0.8, β0 =2, and β=0.7356 connecting the circular part at the point M1 making use of the centre C1, the relative depth of the head would exceed unity, that is rather not recommen- ded. An alternative solution obtained with the use of the points M2 and C2 gives the head meridian shown in the illustration, with the relative depth β=0.7356. This is much better, however, the pattern of the dimensionless equivalent stress shown for the cosinusoidal part of the head is rather unfavourable as it exceeds the level of σ̃0eq = √ 3/2 occurring in the cylindrical part of the vessel. The optimization problem of the head in terms of the variables α1 and β0 is formulated as follows: • optimization criterion min α1β0 {β} Shaping of a middle surface of a dished head... 303 • constraints – the strength condition σ̃eq,max ¬ √ 3 2 Therefore, the final solution to the problem should consist in finding a pair of α1 and b0 parameters so adjusted as to obtain a possibly small β value with the equivalent stress level kept below the value of σ̃0eq = √ 3/2. Numerical analysis has shown that such a solution exists for α1 =1.2212 and β0 =1.6679, which is depicted in Fig.5. Fig. 5. Optimal pattern of the dimensionless equivalent stress for α1 =1.2212, β0 =1.6679, and β=0.7835 TheGaussian curvature 1/(ReRm) of themiddle surface of the head varies significantly and, in consequence, disturbs the membrane state stress. This phenomenonmay be alleviated by increasing the head depth. The MES calculation carried out with the help of the ABAQUS system and shown in Fig.6 confirmed the equivalent stress pattern obtained above for the central line. The computation was performed for an examplary vessel of the radius 1m and shell thickness 10mm. Figure 7 showsagraphical visualization of the equivalent stress at the inner surface of the vessel. It becomes evident that in this case the stress arising at the inner part of the head shell exceeds its level occurring in the cylindrical part. Therefore, another attempt has been undertaken with a view to find a variant so adjusted as to keep the maximum stress at the level characteristic for the cylindrical part. This was possible by enlarging the relative depth of the head. Finally, the relative depth equal to β = 0.8384 gave a satisfactory result depicted in Figs.8 and 9. 304 J. Lewiński, K. Magnucki Fig. 6. Equivalent stresses at the middle, inner and outer surfaces of the head Fig. 7. Visualization of the equivalent stress at the inner surface 4. Conclusions The presented numerical study of the stress state of a cylindrical pressure vessel with convex cosinusoidal-spherical heads enables drawing the following conclusions: • Fulfillment of the condition of continuous curvature in the joint between the head and the cylindrical shell is not sufficient to avoid stress concen- tration in this place. • Further increase in the head depth reduces the value of the concentrated stress. • As a result of shaping the head according to the boundary effect theory, the relative depth β = 0.784 has been obtained for which the stress Shaping of a middle surface of a dished head... 305 Fig. 8. Equivalent stresses at themiddle, inner and outer surfaces of the head for the variant with enlarged relative depth of the head Fig. 9. Visualization of the equivalent stress at the inner surface for the variant with enlarged relative depth of the head concentration should disappear. Nevertheless, a numerical test with the help of FEM has shown the opposite (Fig.6). • Increase in the relative depth up to β = 0.838 eliminates the stress concentration (Fig.8). It should be noticed that the relative depth of standard ellipsoidal heads amounts to the value of β = 0.5 at which a remarkable stress concentration occurs. 306 J. Lewiński, K. Magnucki References 1. Błachut J., Magnucki K., 2008, Strength, stability, and optimization of pressure vessels: Review of selected problems,Applied Mechanics Reviews, 61, 6, 1-33 2. Krivoshapko S.N., 2007, Research on general and axisymmetric ellipsoidal shells used as domes, pressure vessels, and tanks, Applied Mechanics Reviews, 60, 6, 336-355 3. Liu P.-F., Zheng J.-Y., Ma L., Miao C.-J., Wu L.-L., 2008, Calcula- tions of plastic collapse load of pressure vessel using FEA, Journal of Zhejiang University Science A, 9, 7, 900-906 4. 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Conference on Com- putational StructuresTechnology, B.H.V.ToppingandM.Papadrakakis(Edit.), Civil-CompPress, Stirlingshire, Scotland (Paper 138, CD) 11. Yushan Z., Wang Z.R., 1996, Stress analysis of two-arc approximate ellip- soidal pressure vessel heads and parameter optimization, Int. J. Press Vessel Piping, 67, 199-202 12. Yushan Z., Wang Z.R., Wei C., 1996, Stress analysis of approximately ellipsoidal cold-spuntri-arc tankheadandparameteroptimization, Int. J. Press Vessel Piping, 68, 237-242 13. Ventsel E., Krauthammer T., 2001,Thin plates and shells. Theory, ana- lysis and applications, Marcel Dekker, Inc, NewYork, Basel Shaping of a middle surface of a dished head... 307 Kształtowanie środkowej powierzchni wypukłego dna walcowego zbiornika ciśnieniowego Streszczenie Praca dotyczy dna wypukłego walcowego zbiornika ciśnieniowego obciążonego równomiernym ciśnieniem wewnętrznym. Zamieszczono krótki przegląd problemu optymalnego projektowania zbiornika i jego dna. Przedstawiono problem kształto- wania środkowej powierzchni wypukłego dna z zastosowaniem szeregu trygonome- trycznego.Warunek ograniczający kształt południka dna dotyczy ciągłości krzywizn w miejscu połączenia z powłoką walcową. Poszukiwano rozwiązania optymalnego, w którym głębokość dna jest minimalna przy ograniczeniu warunkiem wytrzyma- łości. Wyniki numerycznych obliczeń optymalnych kształtów dna przedstawiono na rysunkach. Manuscript received May 6, 2009; accepted for print July 9, 2009