Appendix FORCES AND DEFORl\'IATIONS IN CIRCULAR RINGS Shells of revolution are frequently connected with circular rings, to which they transmit forces and moments. The theory of stresses and ,deformations of such rings is part of the theory of structures. While a. few formulas are found almost everywhere, it is not easy to find the complete set in books of this kind. Therefore they have been compiled here. In all formulas we assume that the ring is thin, i.e. that the di­ mensions of its cross section are small compared with the radius. The axis of the ring is supposed to pass through the centroids of all cross sections. One principal axis of these sections lies in the plane of the ring. Besides those explained in the figures, the following notations have been used: A = area of cross section, 11 = moment of inertia for the centroidal axis in the plane of the ring, I 2 = moment of inertia for the centroida.l axis normal to the plane of the ring, J T = torsional rigidity factor of the section. Where moments and angular displacements have been represented by arrows, the corkscrew rule applies for the interpretation. 1. Radial Load (Fig. A.l) The load per unit length of the axis is assumed to be p = Pn cosnO. Stress resultants: Pna2 N =- ~·a cos nO M =- - --cos nO. n·- 1 ' 2 n 2 - 1 Displacements: 508 APPENDIX These formulas are not valid for n = 0 and n = 1. For n = 0 (axisym­ metric load) they must be replaced by the well-known formulas pa" N=pa, V= 0, W=EA. (a) (b) (c) J<'i!(. A.!. Radial loud ]'or n = 1 the problem does not exist, since a load of this type is. not self-equilibrating. 2. 'l'angt•ntial J,oau (Fig. A .2) The load per unit length of the axis is assumed to be p = p, sinnO. Stress resultants: np.a O p.a" N cos n , = n·--.--1 .il1 2 = n (n" _ 1) cos nO. Fig. A.2. Tangential load CIRCULAR Rl~GS 509 Displacements : Pna' W=- n(n2-1)2E[2 cosn8. For n = 0 and n = 1 this problem does not exiHt, because the ex­ ternal forces would not be in equilibrium. 3. Load Normal to the Plane of the Ring (Fig. A.3) The load per unit length of the axis is assumed to be p = Pn cosn8. Stress resultants : l,r p,. a2 LJ Pnft'1. • ~u = -.-- cos n u , =- ( ,, i) 1 n-- 1 1rfr n n-- smn(J, Displacements: deflection U= (n;~:) 2 b/1~ + n 2 ~JJ cosn8, rotation of the cross section in its plane Pn a 3 [ 1 , 1 1 tp=(n2-1)2 F:"i~'GJr cosne. Fig. A.3. Loau normal to the plane of the ring For n = 0 and n = 1 this problem does not exist, since the loads would not be self-equilibrating. 4. }~xtcrnal IUoments, Turning about the Ring Axis (Fig. AA) The couple applied per unit length of the axis of the ring is assumed to be m= mn cosne. 510 APPENDIX Stress resultants: m.a () ~,r nm,.a . () J}f1 = --.-- cosn , 1r11' = -.-- sm n . n·- 1 n·- 1 Displacements: Fig. A.4. )Ioment load The case n = 1 does not exist, since this special load would not be in equilibrium. In the case n = 0 the formulas for the moments yield correctly i.e. we are dealing with pure bending. Also the formula for 1p is correct and yields ma2 1jJ = E I~ ' but the displacement u, a rigid-body movement normal to the plane of the ring, may assume any desired value. In practical problems the loads are usually not applied at the ring axis but at some other circular fiber. It is then necessary to replace them by an equivalent set of loads of the kind used in the preceding formulas. This substitution must be done with some care. BIBLIOGRAPHY In their monograph on the stability of circular cylindrical shells (see p. 515), GRIGOLYUK and KABANOV present a graph of the number of papers that have appeared per year, on this special subject. Somewhere near 1970, the extrapolated curve appears to have a vertical asymptote. A similar graph might be drawn for the entire shell literature. Although in no case will the number of papers ever reach infinity, it has grown so tremendously that it would be a hopeless attempt to com­ pile a bibliography of even the more important works, the more so as each follow­ ing year is likely to bring another flood. On the other hand, many of the funda­ mental publications of the first decades of this century have appeared at places which are no longer readily accessible and are fading from view. Therefore, this Bibliography lists only a few of the older papers, which have been stepping stones in the development of the theory, and some books and papers of the last decades, which were considered particularly helpful for further study. No attempt at com­ pleteness has been made and no attempt at outlining the history of shell theory. For the great mass of modern literature, including that which is still to appear in future years, the reader is referred to Applied ~lechanics Reviews. Fundamentals of Shell Theory, Text Books The first major presentation of shell theory is that of A. E. H. LovE: .\Treatise on the Mathematical Theory of Elasticity, 4th ed., Cambridge 1927, chapters 2:3-24. His set of general shell equations has served several generations of authors as the starting point for their work. The first book devoted exclusively to shell theory is W. FLUGGE: Statik und Dynamik der Schalen, 3rd ed., Berlin 1962. This book, like the present one, does not derive general equations and then specialize them, but considers different classes of shells and develops the basic equations for each of them separately, but using the same fundamental principles. This has the advan­ tage of simplicity, but does not cover exotic shapes. In the same style the follow­ ing two books are written: S. TIMOSHENKO, S. WorNOWSKY-KRIEGER: Theory of Plates and Shells, 2nd ed., New York 1959, pp. 429-568; K. GIR!OIANN: Fliichen­ tragwerke, 5th ed., Wien 1959, pp. 352-582. Both books contain other subjects beside shell theory. GIRK.'r!ANN's book has been written for civil engineers and is restricted to problems pertaining to this special field of application. The book by H. KRAus: Thin Elastic Shells, New York 1967, derives general equations referred to lines of principal curvature and then specializes to the exclusive treatment of shells of revolution. Books containing general equations similar to those of LovE have appeared in Russia: V. Z. VLASSOV: General Theory of Shells (in Russian), Moscow 1949, German translation: Allgemeine Schalentheorie und ihre Anwendung in der Technik, by A. KROMM, Berlin 1958; V. V. NovozHILOV: The Theory of Thin Shells, English translation by P. G. LowE and J. R. M. RADOK, Groningen 1959; A. L. GoL'DENYEIZER: Theory of Elastic Thin Shells, English translation by G. HERRMANN, New York 1961. ~With the exception of GoL'DENVEIZER, all these 512 BIBLIOGRAPHY authors use lines of principal curvature as coordinate lines. This impairs the use of these equations for such simple problems as the paraboloid shells with rectangular boundary. The use of absolutely general coordinates calls for the methods of tensor analysis. Work of this kind is found in the book by A. E. GREEN, W. ZERNA: Theoretical Elasticity, Oxford 1954, chapters 10-14; in papers by P. M. NAGHD!: Foundations of elastic shell theory, Progress in Solid 11ech. -l (196:1), 1-90; A new derivation of the equations of elastic shells, Intern. J. Eng. Sci. 1 (1963), 509-522; and in the book by W. FLtjGGE: Tensor Analysis and Continuum 1Iechanics, Ber­ lin 1971, chapter 9. Jiembrane Theory of Shells In recent years not many publications on this subject have appeared. It seems that the needs of the daily work are covered by the research of earlier decades and that the fallacies of membrane solutions as explained at many places in this book have detracted the attention from their basic usefulness in guiding the designer toward those shell structures which support their load essentially with membrane forces and, therefore, are stiffer and cheaper than others. It may suffice here to list a few of the earlier papers, which represent essential steps toward the understand­ ing of the mechanics of membrane stress systems. Already in the 19th century formulas for pressure vessels and water tanks were known and widely ttsed. The membrane theory of shells of revolution under an­ symmetric loads begins with a paper by H. REISSNER: Spannungen in Kugelschalen (Kuppeln), MuLU:R-BRESLAU .Festschrift, Leipzig 1912, pp. 181-19:3. The discussion of the consequences of negative GAussian curvature was opened by \V. FLfGGE: Zur 1Iembrantheorie der Drehschalen negativer Kriimmung, Z. angew. :Math. Mech. ~;;;~i (1947), 65-70. A critical presentation of the membrane theory of shells of revolution was given by C. TRUESDELL: The membrane theory of shells of revo­ lution, Trans. Am. 1\Iath. Soc. ;)S (1945), 96-166. Cylindrical shells under general load were first studied by D. Tuo~rA: Die Beanspruchung freitragender gefiillter 1-~ohrc durch das Gewicht der Fliissigkeit, Z. ges. Turbinenwesen 17 (1920), 49-52. The usefulness of AIRY's stress function for the solution of membrane shell problems was discovered by A. PUCHER: t.Jber den Spannungszustand in gekriimmten Fliichen, Beton u. Eisen :J:l ( 19:34), 298-:304. Paraboloids of negative GAussian curvature, still without the benefit of AIRY's stress function, were studied by B.