UNCLASSIFIED,AD 242 194,ARMED SERVICES TECHNICAL INFORMATION AGENCY. ARLINGTON HALL STATION,ARLINGTON 12 VIRGINIA,UNCILASSIFIED. NOTICE When government or other drawings speci,fications or other data are used for any purpose. other than in connection with a definitely related. government procurement operation the U S, Government thereby incurs no responsibility nor any. obligation whatsoever and the fact that the Govern. ment may have formilated furnished or in any way, supplied the said drawings specifications or other. data is not to be regarded by implication or other. wise as in any manner licensing the holder or any, other person or corporation or conveying any rights. or permission to manufacture use or sell any,patented invention that may in any way be related. OFFICE OF NAVAL RESEARCH,Contract Nonr 562 10,NR 064 406. Technical Report No 63, DESIGN OF THIN WALLED TORISPHERICAL AND TORICONICAL PRESSURE. VESSEL HEADS,R T Shield and D C Drucker,DIVISION OF APPLIED MATHEMATICS. BROWN UNIVERSITY,PROVIDENCE R I,C 11 63 July 1960,DESIGN OF THIN WALLED TORISPHERICAL AND. TORICONICAL PRESSURE VESSEL HEADS,by R T Shield and D C Drucker. The failure under hydrostatic test of a large storage. vessel designed in accordance with current practice st mulated. earlier analytical studies0 This paper gives curves and a. table useful for the design and analysis of the knuckle region. of a thin torispherical or toriconical head of an unfired. cylindrical vessel A simple but surprisingly adequate approxi. mate formula is presented for the limit pressure npD at whilch. appreciable plastic deformations occur,noD 0 33 5 5 Z D 28 I 2 2 r 0 0006. where pD is the design pressure d 0 is the yield stress of the. material and n is the factor of safety The thickness t of. the knuckle region is assumed uniform Upper and lower bound. calculations were made for ratios of knuckle radius r to. cylinder diameter D of 0o06 0 08 0 10 0 12 0 142 and 0 16. and ratios of spherical cap radius L to D of 1 0 0 9 0 8. The results presented in this paper were obtained in the. course of research sponsored by the Office of Naval Research. under Contract Nonr 562 10 with Brown University,Professor of Applied Mathematics Brown University. Professor of Engineering Brown University, 0 7 and 0 6 Toriconical heads may be designed or analyzed. closely enough by interpreting yo of the Table as the complement. of the half angle of the cone,Introduction,The design of pressure vessels requires the long. experience distilled into the ASME Code to avoid overlooking. many important factors In principle the most straightforward. of the difficult problems is the design of an unreinforced. knuckle region of uniform thickness in an unfired pressure. vessel subjected to interior pressure This topic is discussed. at length in the Code and it might well be expected that little. remained to be resolved Surprisingly analytical studiesl12. stimulated by reports of a failure under hydrostatic test. demonstrated conclusively that the thickness required by the. Code is inadequate for a range of designs This range is one. of small pressures and consequently of vessels whose wall. G D Galletly has studied elastic behavior in Tori spherIcal. Shells A Caution to Designers 9 Journal of Engineering. for Industry ASME v 81 1959 pp 51 62 and On, Particular Integrals for Toroidal Shells Subjected to. Uniform Internal Pressure Journal of Applied Mechanics2. ASME v 25 1958 pp 412 4139, 2 D C Drucker and R T Shield have studied plastic behavior. in Limit Strength of Thin Walled Pressure Vessels with an. ASME Standard Torispherical Head Proceedings Third U S. National Congress of Applied Mechanics Brown University. 1958 ASME pp 665 672 and Limit Analysis of Symmetrically. Loaded Thin Shells of Revolution Journal of Applied. Mechanics ASME v 26 1959 pp 61 68a, thickness is small compared with the knuckle radius as well as. the radius of the vessel itself It did not in all likelihood. engage the serious attention of the framers of the Code who were. concerned primarily with pressures exceeding several hundred. pounds per square inch At these higher pressures a sharply. curved knuckle would have a radius which is not very large. compared with the wall thickness and so the knuckle would not. be flexible and weak, A design of adequate strength must provide a reasonable. factor of safety against reaching the limit pressure the pres. sure at which significantly large plastic deformation will take. place Many additional practical matters as well must be taken. into account in the designo Among these are corrosion allowance. thinning allowance and joint efficiency They will not be. considered here except by implication in the designation of the. limit pressure as npD where n is a factor of safety and pD is. the design or working pressure, The limit pressure is especially significant in a cold. environment for those steels which are prone to brittle fracture. Appreciable plastic deformation below the transition temperature. is almost certain to initiate a brittle fracture Above this. rather ill defined transition temperature the shape of a vessel. of ductile material will be able to change sufficiently to carry. the pressure without catastrophic failure The pressure simply. cold forms the head to a quite different but much better shape. for containing pressure, A Qualitative piscussion Qf the Behavior of Presulre Vessels. A thin walled vessel under interior pressure is most. efficient when it can carry the pressure as a membrane in. biaxial tension However the shape required for this desirabls. membrane behavior has a height of head H 0 26D which often. appears too large from the fabrication or space utilization. point of view Torispherical heads are employed to reduce H. appreciably but they cannot act in biaxial tensiono they must. carry circumferential compression in the knuckle and also resist. bending Their load carrying capacity as pure membranes no. moment resistance shown in the Table as pMD 2 dot and plotted. on some of the graphs at t D 0 is extremely low Actually. a very thin shell acting as a membrane would buckle in circum. ferential compression,As the pressure builds up it tends to force the. spherical cap outward along the axis and the meridional membrane. tensions pull the toroidal knuckle inward toward the axis If. the torus wall is thick enough to avoid buckling but thin com. pared with the radius of the knuckle and the material does not. work harden a plastic hinge circle will form at B Fig 1 to. permit the central region of the knuckle to compress in the. circumferential direction and bend inwards A hinge circle. will form at C in the spherical cap and the third hinge circle. A usually forms in the cylinder The entire knuckle region. 3 R A Struble Biezeno Pressure Vessel Heads JournIal of. Applied Mechanics ASME v 23 1956 pp 642 A4 5c, between A and C is plastic because inward motion of appreciable. extent means plastic contraction of the circumference A thin. walled sharply curved knuckle region is far weaker than the. main part of the spherical cap or the cylindrical portion of. the vessel On the other hand if the torus wall is not so. thin compared with the knuckle radius the knuckle region is. stiff and strong and acts somewhat like a stiffening ring at. the junction of a spherical cap and a cylinder The ASME Code. which requires very little variation of npDD 50t with tD. apparently contains the implicit assumption that ordinarily. the resistance to inward motion of the knuckle region is. adequately high Although true for vessels designed to carry. large pressure the assumption is not valid for many sttorage. vessels and other low pressure containers For these thin. walled vessels there is a large variation of the value of. np DD 2ot with t D as shown in Figs 2 5 On the other h ndq the. dotted lines for values of npDD 25o t greater than unity show. that for less sharply curved knuckles and for relatively thic 1. knuckles the knuckle region is stronger than the main cylindri. cal part of the vessel,Design Curves and Formula, The upper and lower bound theorems of limit anaylsis. and design were used to calculate the limit pressure Therefore. even within the usual idealizatibns of the theory of plast1cityg. 4 D C Drucker W Prager and H J Greenberg I Extended. Limit Design Theorems for Continuous Media Quarterly cf. Applied Mathematics v 9 1952 pp 381 389, the exact answer is bounded rather than determined directly. Curves are plotted in Figs 2 and 3 for p UD 2dot the upper. unsafe values computed for np DD 2d t and in Figs 4 and. for p1D 2d0 t the lower oversafe values The designer then. can make an independent judgment of the appropriate values to. However if moderate accuracy is good enough or if a. preliminary design is sought Fig 6 should prove a very helpfuol. alternative An approximate plot of t D vs H b for discrete. values of npD d 0o it gives a clear picture of the penalty to. be paid for the advantage of decreasing the axial length of. the vessel The agreement with the mean of the upper and lower. bound calculations also shown on Fig 6 varies with r b and. L D but to a much smaller extent than might be expectedo. Remarkably good agreement with the limit calculations. can be achieved through use of the variable t L which is of. prime importance in the ASME Code The excellent fit of the. simple formula,0 33 5 5 WDL 28 i 2 2 0 0006, is illustrated in Fig 7 a plot of t L vs npD 6o for two values. of r b The relatively minor variation with L b is also a. feature of the Code However the Code calls for a linear. variation of t L with increasing pressure and there is no way. of adjusting a straight line to the proper curves without. being unsafe or far too safe The lack of safety is all too. evident in Fig 8 a plot of the formula for discrete values of. r b which permits the designer to select t L for a given pres. sure or to check the pressure carrying capacity of an existing. design Again the designer is urged to return to Figs 2 5 to. obtain upper and lower bounds on his factor of safety if he is. forced to design with a very small margin, The Appendix contains detailed information on the basis. and the methods of calculation of Figs 2 5o It supplements. the discussion contained in the earlier papers and is not. complete in itself In essence the Tresca or maximum shearing. stress criterion of yield is employed and the yield surface. for the shell is a cut off parabolic approximation to the exact. shape for a symmetrically loaded cylindrical shell. Toriconical Heads, The values of t L and npD do plotted for a given torus. apply equally well to torispherical and to toriconical heads. The Table can be used to obtain the appropriate interpolated. value of L D for Figs 2 5 if desired The angle yo is the. complement of the torus angle and therefore the complement of. the half angle of the cone, The equations of equilibrium for the various portions. of the vessel cylinder torus and sphere are given in the. references of footnote 2 The term involving the circimferen. tial bending moment Me is omitted from the equations of. equilibrium for the torus and the sphere as M has little. influence in carrying load for thin shells at sections not too. near the axis of symmetry The meridional bending moment MN. is similarly omitted but its derivative is retained0. As Me is considered as a passive moment in the curved. portions of the shell as well as in the cylinder full use of. M and the meridional and circumferential force resultants NIP. and N in carrying the internal pressure p is obtained by. using the yield condition on N No M for the cylinder In. order to approximate to this yield condition or surface the. circumscribing surface consisting of a parabolic cylinder with. four cut off planes is used2 In the region of interest. between the hinge circles A B C of Fig l NX is tensile and. N is compressive For this region to be at yield the parabolic. prism yield surface requires,Nq N ot IM1 0t,o21 NWdot 2 3 1. It is assumed that at the hinge circles A and C in the cylinder. and the sphere M attains its largest negative value and at. hinge circle B in the torus M attains its largest positive. value The shear force Q is zero at the hinge circles Under. these conditions the equations of equilibrium can be integrated. to provide the distribution of Ne N M and Q in the plastic. It is found that in the cylinder,1l2t D 2 2,M C5 t l k t I 1do i 2 DD x 2. 1P r 0 o2D 2 dt 0Xt,D 0 P xo x, where x measures distance from the junction with the torus and. xo defines the location of the hinge circle A In the torus. M I t 2 R r sinpm,D D PR 1 cos Y Y,rd 0 t 2 02d,t D2sin29m 2d t singm. cossq k y k 9 log R r ssin n,0 2t t sin 9m r sin cp k y k m A. 2 0tsiny Rm,w r R tanI Y,k cp R rRdQsin9 2R 1 2 an lf n2I I 6. 0 L r 2 R2 2o1 2, D is the angle between the meridional normal and the axis of. the shell and Im is the location of the hinge circle B In thf e. sphere with the assumption that y yo is small2,L 0 t L 26 ot Ys 7. where ys defines the location of the hinge circle Co. The four quantities p ym ys and xo are determined from. the conditions that M and Q are continuous at the junctions. of the cylinder and torus x O p n 2 and the torus and sphere. Y y0o These conditions can be written,D 2 J Pm 2D,d t2 a m o b ym 9. C Tm D d 9m,9 os U pm 2Dot e m 2d t,o s s S g ym 2DD h ym 12. where the functions not previously defined are given by. rR 1 sin TM,a cp 2 sin pm 13,b p 2 X1 lg R r t 1,m D R r sinym D. C cm Rcotq 15,d Ym Ilk 9 2 k 16,1 cos M 90 I,e cpm 2 sincm 7. f m c 2 Cos ok m 2 lOgR T17,g m R sin 19,hm D sin km. h ym RsinTO k cpm 20,j c m 1 t R r sinym 21,m 2 1DD D2 sin2cpm. Equations 9 12 were solved for pD 2dot m cps and D for. given values of the parameters t D L D and r D which define. the geometry of the vessel The following values of the para. meters were used,t D 0 002 0 00o 0 006 8 0 010 0 0129 0 0141. L D 1 0 09 M8 0 7 0 6,r D 0 06 0 08 O0lO 0 12 0 11 0 16. In the numerical method used a trial value amwas chosen for. qm and the functions of ym occurring on the right hand sides. of equations 9 12 were evaluated By elimination of. To Qs between 11 and 12 a quadratic equation was obtained. for pD 2 0t The positive root of this equation was then. substituted in 9 and 10 to give two values of xo D. The difference between these two values was evaluated and the. procedure was repeated with another trial value ymfor cp. and again the difference between the two values of x D 2 was. found Linear interpolation between a and b was then. used to give a better approximation to the true value of. mP The process was repeated until the magnitude of the. difference between the two values of x D 2 as provided by.

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