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. LA~'FAILLE: .:\le• moire sur !'etude generale des surfaces gauches minces, Intern. Assoc. Bridge Struct. Engg., Pub!. :l (1935), 295-332; and F. ADIOND: Etude statique des voiles minces en parabolo!de hyperbolique travaillant sans ftexion, Intern. Assoc. Bridge ~truct. Engg., Pub!. -l (19:l6), 1-112. The concept of the polygonal dome is due to F. DrsCHINGER: Theorie der Vieleckskuppeln und der Zusammenhang m it den ein• beschriebenen Rotationsschalen, Beton u. Eisen ~S (1929), 100-107, 119-122, 150 -156, 169-175. The solution for unsymmetric loads \vas given by W. FLUGGE: Statik und Dynamik der Scha I en, 1st ed., Berlin 19:!4, pp. 85-91. The theory of affine shells dates back to another paper by F. DISCHTNGER: Der Spannungszustand in affinen Schalen und Raumfachwerken unter Anwendung des statischen l\Iassen• ausgleichs, Bauing. lj (1936), 228-231, 260-267, 289-295. He finds the essential facts without using AIRY's stress function. The concept of the folded plate structure was introduced by H. CRAE:IIER: All• gemeine Theorie der Faltwerke, Beton u. Eisen :!9 (1930), 276-281; and G. EnLERS: Die Spannungsermittlung in Flachentragwerken, Beton u. Eisen :!9 (1930), 281-286, 291-296. The theory of the inextensional deformation of curved surfaces may be foun BIBLIOGltAPHY 513
in every text on differential geometry. For shells of revolution, the harmonic cmn• ponents of this deformation have been used as approximations to the vibration modes, see Lord RAYLEIGH: Theory of Sound, vol. 1, 2nd ed., London 1894, p. 402; M. J. 0. STRUTT: Eigenschwingungen einer Kegelschale, Ann. Phys. V, 17 (19:33), 729-735. This use of the inextensional deformations is, of course, only possible if the shell is so supported that these deformations can develop freely. The membrane deformation of shells of arbitrary shape has been studied by F. T. GEYLING in his Stanford thesis (seep. 518), see also W. FLUGGE, F. T. GEY· LING: A general theory of deformations of membrane shells, Proc. 9th Intern. Congr. Appl. 11ech., Bruxelles 1956, vol. 6, pp. 250-262 and Intern. Assoc. Bridge Struct. Engg., Pub!. 17 (1957), 2:!-46. The basic equations have also been developed by E. BoLCSKEI: DCformation des voiles minces, Acta Techn. Acad. Sci. Hungaricae ;; (1952), 489-506.
Bending 'l'hcory of Shells
The elastic law (5.9) and the differential equations (5.13) of the cylindrical shell were derived in this form by W. FLUGGE: Die Stabilitat der Kreiszylinderschale, Ing.-Arch. 3 ( 1932), 463-506. The underlying idea, namely the strict adherence to KIRCHHOFF's hypothesis of the conservation of normals, has later been applied by many authors to other shells, see also (6.5) of this book. It has also been used as the basis of very general shell equations, see the papers by P . .M. NAGHD! and the book by W. FLUGGE mentioned on p. 512. Because of the bulkiness of these equations many efforts have been made at obtaining a simpler set of shell equations. The most radical attempt of this kind, underlying (5.12) and (5.18), has been made by L. H. DoNNELL: Stability of thin• walled tubes under torsion, N ACA, Rep. 4 79 ( 193:3). Also this line of thought has been applied to shells of various shapes. Since DoNNELL's equations are not con• sistent and may, in certain cases, lead to serious errors, while (5.13) carry termR which, in most cases, are insignificant, many authors have suggested intermediate formulations. The discussion of this subject is still in progress. The solution of the inhomogeneous cylinder problem (Section 5.2) was given by H. ltEISSNER: Formiinderungen und Spannungen einer diinnwandigen, an den Riin• dern frei aufliegenden Zylinderschale, Z. angew. l\Iath. Mech. 13 (1923), 1:33-1:!8; and the principal ideas of Section 5.:3 have been taken from K. )l!ESEL: Dber die Festigkeit von Kreiszylinderschalen mit nicht-achsensymmetrischer Belastung, Ing-Arch.l (1930), 22-71. This paper also gives an approximate solution omitting the weakly damped part. The opposite approximation, which drops the strongly damped part of the solution, has been treated by E. GRUBER: Die Berechnung zy• lindrischer, biegnngssteifer Schalen unter beliebigem Lastangriff. Intern. Assoc. Bridge Struct. Engg., Pub!. 2 (1934), 196-204;·H. WAGNER, H. SmoN: Dber die Krafteinleitung in diinnwandige Zylinderschalen, Luftf.-Forschg. 13 (1936), 293 -:~08. The boundary conditions of a plane plate have been formulated by G. KIRCHHOFF: Vber das Gleichgewicht und die Bewegungen einer elastis0hen Scheibe, .J. reine angew. Math. -!0 (1850), 51-88. For the corresponding shell problem see A. B. BASSETT: On the extension and flexure of cylindrical and spherical thin elastic shells, Phi!. Trans. ltoy. Soc. London A, 181 (1890), 433-480. The barrel vault theory goes back to U. FrNSTERWALDER: Die querversteiften zylindrischen Scha• lengewiilbe mit kreisfiirmigem Querschnitt, Ing.-Arch. 4 (1933), 43-65. The simpli• fied barrel vault theory of Section 5.4.3 has been created by H. ScHORER: Line load action in thin cylindrical shells, Proc. Am. Soc. Civ. Eng. 61 (1935), 281-316. A completely different approach has been made by H. LuNDGREN: Cylindrical Shells,
Fliigge, i'ltresses in Shell•. 2nd Eu. 514 BIBLIOGRAPHY vol. 1, Cylindrical Roofs, Copenhagen 1951. Since the actual stresses in a barrel vault differ so thoroughly from the membrane stresses, the author considers the shell inclusive the edge members as a simple beam and calculates Oz from the straight-line law. With the result he goes successively through all the shell equations and obtains at last a correction to oz. The iteration cycle may be repeated as often as needed. The practical importance of the barrel vault problem has led to a number of books devoted more or less exclusively to this special subject, often with extended numerical tables. We mention the following: R. S. JENKINS: Theory and Design of Cylindrical Shell Structures, Modern Building Techniques, Bull. 1, London 194 7; J. E. GtBSON, D. W. COOPER: The Design of Cylindrical Shell Roofs, New York 1954; R. RABICH: Randwerttabellen zur Berechnung von Kreiszylinderschalen, Dresden 1954; and Manual No. 31 of the American Society of Civil Engineers: Design of Cylindrical Shell Roofs, 2nd ed., New York 1952. The most valuable part of this book, its numerical tables, has been reprinted in D. P. BILLINGTON: Thin Shell Concrete Structures, New York 1965. In the theory of shells of revolution under axisymmetric load, the decisive step was the introduction of Q+ and X as unknowns by H. REISSNEB. in his paper in the }ItiLLE:a-B:&ESLAU Festschrift, see p. 512. The idea has been much extended by E. ::\IEISSNEB.: Das Elastizitii.tsproblem fiir diinne Schalen von Ringflii.chen-, Kugel• und Kegelform, Phys. Z. U (1913), 343-349; Uber Elastizitii.t und Festigkeit diin• ner Schalen, Vj.-Schr. Naturf. Ges. Ziirich t~O (1915), 23--47; Zur Elastizitii.t diin• ner Schalen, Atti Congr. Intern. Mat., Bologna 1928, vol. 5, pp. 155-158. The idea of asymptotic integration was introduced by 0. BLU:\IENTH.\L: Uber die asymptoti• sche Integration von Differentialgleichungen mit Anwendung auf die Berechnung von Spannungen in Kugelschalen, Z. Math. Phys. 62 (1914), 34:1-358. The highly simplified (and very popular) solution given in Section 6.2.1.4 is due to J. W. GECKE• LER: Uber die Festigkeit achsensymmetrischer Schalen, Forschg.-Arb. Ing.• wes., vol. 276, Berlin 1926. In another paper: Zur Theorie der Elastizitii.t flacher rotationssymmetrischer Schalen, lng.-Arch. 1 (1930), 255-270, the same author gave the solution presented in Sections 6.2.1.5 and 6.2.1.6. The mathematical difficulties of the bending theory of shells may be reduced substantially by restricting the discussion to shells which are almost plane plates. Earlier papers on such shallow shells were mainly concerned with a nonlinear buck• ling phenomenon. A general theory of shallow shells was formulated by K. ::\[AR• HUERRE: Zur Theorie der gekriimmten Platte groBer Formii.nderung, Proc. 5th Intern. Congr. Appl. 1\'Ieeh., Cambridge, Mass. 1939, pp. 93-101. The theory ha11 been further developed and applied to many problems in various papers by E. REJSS• NKR. The thermal stress singularities treated in Section 7.3 were first discussed in a paper by W. FLUGGE, D. A. CoN:&AD: Thermal singularities for cylindrical shells, Proc. :kd US Nat. Congr. Appl. 1\'Iech., Providence, R. I. 1958, pp. 321-328. Shells of revolution under nonaxisymmetric edge"loads have been dealt with in the following papers: A. HA YEB.S: Asymptotische Biegetheorie der unbelasteten Kugelschale, Ing.-Arch. 6 (1935), 282-213; H. NoLr.Au: Der Spannungszustand der biegungssteifen Kegelschale mit linear veriinderlicher Wandstiirke unter beliebigcr Belastung, Z. angew. Math. Mech. 24 (1944), 10-34; W. H. WITTB.ICK: Edge stresses in thin shells of revolution, Australian J. Appl. Sci. 8 (1957), 235-260; and in the Stanford theses of F. A. LECKIE and C. R. STEELE (see p. 518). In the last two decades the theory of plasticity has been applied to shells. The state of the art, including references to many papers, has been presented in the book by P. G. HoDGE: Limit Analysis of Rotationally Symmetric Plates and Shells, Englewood Cliffs, N. J. 1963. A complete solution for shells of revolution under axisymmetri<' load, based on T:&ESCA's yield condition, has been attempted in the BIBLIOGRAPHY 515 following papers: W. FLUGGE, T. NAKAMURA: Plastic analysis of shells of revolu• tion under axisymmetric loads, Ing.-Arch. :l! (1965), 238-247; W. FLUGGE, J. C. GERDEEN: Axisymmetric plastic collapse of shells of revolution according to the NAKAMURA yield condition, Proc. 12th Intern. Congr. Appl. Mech., Stanford 1068, pp. 209-220.
Buckling of Shells
The elastic stability of cylindrical shells under axial compression, external pres• sure, and the combined action of both, has been studied by several authors begin• ning 1911. The presentation in this book follows the paper by W. FLUGGE: Die Sta• bilitat der Kreiszylinderschale, Ing.-Arch. 3 (1932), 463-506. The buckling under shear load has been studied by E. ScHWERIN: Die Torsion des diinnwandigen Roh• res, Z. angew. Math. l\'Iech. ;) (1925), 235-253; L. H. DONNELL: Stability of thin• walled tubes under torsion, NACA, Rep. 479 (1933). The combination of shear with axial and circumferential compression is the subject of the following papers by A. KROMJ\I: Die Stabilitatsgrenze eines gekriimmten Plattenstreifens bei Be• anspruchung durch Schub- und Langskrafte, Luftf.-Forschg. liJ ( 1938), 517-526; Knickfestigkeit gekriimmter Plattenstreifen unter Schub- und Druckkraften, Jb. deutsch. Luftf.-Forschg. 1940, vol. 1, pp. 8:32-840; Beulfestigkeit von versteiften Zylinderschalen mit Schub und Innendruck, Jb. deutsch. Luftf.-Forschg. 1942, vol. 1, pp. 596-601; Die Stabilitatsgrenze der Kreiszylinderschale bei Beanspruchung durch Schub- und Liingskriifte, Jb. deutsch. Luftf.-Forschg. 1942, vol. 1, pp. 60i -616. Nonuniform axial compression has been treated by W. FLUGUE: Ing.-Arch. 3 (see above). The cylinder shown in .I!'ig. 3.7, carrying a simulated dead load (with a sine distribution in the x direction) has been studied in the Stanford thesis of G. C.-:\I. CHIANG. Also a shell subjected to axisymmetric bending stresses may have a genuine buckling problem, see the Stanford thesis of H. V. HAHNE. Beam-column problems have been investigated by L. FoPl'L: .Achsensymmetri• sches Ausknicken zylindrischer ~chalen, S.-Ber. Bayr. .Akad. Wiss. 1926, 27-40; J. W. GECKELER: Plastisches Knicken der Wandung von Hohlzylindern und einige andere Faltungserscheinungen an Schalen und Blechen, Z. angew. Math. l\Iech. S (1928), 341-352. Both authors restricted the theory to axisymmetric deformation. The linear theory of general imperfections was given by W. FLUGGE, Ing.-Arch. 3 (see above). The nonlinear theory of cylinder buckling started with the papers by L. H. DoNNELL: A new theory for the buckling of thin cylinders under axial compression and bending, Trans . .Am. Soc. Mech. Eng. ;)6 (1934), 795-806; T.v. KAR)IAN, H. S. TsrEN: The buckling of thin cylindrical shells under axial compression, J. Aeron. Sci. 8 (1941), 303-312; H. S. TsrEN: A theory for the buckling of thin shells, J. Aeron. Sci. 9 (1942), 373-384. In the last decades this problem (mostly restricted to axial compression) has produced a voluminous literature and is likely to produce more. A comprehensive presentation of the field of cylinder buckling including the experimental evidence may be found in the book by E. I. GRIGOLYUK, V. V. KA• BANOV: Stability of Circular Cylindrical Shells (in Russian), Moscow 1969. This book also contains an extensive bibliography. Another nonlinear problem occurs when a long, thin-walled tube is subjected to pure bending. The circular section assumes an oval shape and the bending moment increases less than proportional to the deflection, reaches a maximum, and then decreases. The maximum defines the collapse load of the shell. This problem was 33* 516 BIBLIOGRAPHY
studied first by L. G. BRAZIER: On the tiexure of thin cylindrical shells and other thin sections, Proc. Roy. Soc. London A, 116 (1927), 104-l14. Rectangular cylindrical panels surrounded by stiffeners do not collapse as read• ily as an EuLER column when the buckling load has been reached. As with flat plates, there exists a problem of post-buckling behavior. The following papers may be consulted about this nonlinear phenomenon: T. E. ScHUNCK: Der zylindrische Schalenstreifen oberhalb der Beulgrenze, Ing.-Arch. 16 (1948), 403-432; D. A. M. LEGGETT, R. P. N. JoNES: The behaviour of a cylindrical shell under axial com• pression when the buckling load has been exceeded, Aeron. RtJ::s. Comm., Rep . .i\Iem. 2190 (1942); H. L. LANGHAAR, A. P. BoRESI: Buckling and post-buckling behavior of a cylindrical shell subjected to external pressure, Univ. of Illinois, 1956; H. F. l\IrcHIELSEN: The behavior of thin cylindrical shells after buckling under axial compression, J. Aeron. Sci. 1;) (1948), 738-744; J. KEMPNER: Post• buckling behavior of axially compressed circular cylindrical shells, J. Aeron. Sci. 21 (1954), 329-:J35. The stability of the spherical shell under external pressure has been studied by R. ZoELLY: tlber ein Knickungsproblem an der Kugelschale. Diss. Ziirich 1!H;3; E. ScHWERIN: Zur Stabilitat der diinnwandigen Hohlkugel unter gleichmaBigtJm AuBendruck, Z. angew. :Math. :Mech. 2 (1922), 81-91. Both authors consider only axisymmetric deformations. The complete solution of the problem was given by A. v. D. N~]UT: The elastic stability of the thin-walled sphere (in Dutch), Diss. Delft 19:!2. A shallow spherical cap with load acting on its convex side presents a nonlinear ,;tability problem - the snap-through problem. Among the earlier papers on the subject are the following: C. B. BIEZENO: O"ber die Bestimmung der Durchschlags• kraft einer schwach gekriimmten kreisfOrmigen Platte, Z. 11ngew. :\lath. lHech. l;) (19:35), 10-22; T.v. K..\&:vrA.N, H. S. TsiEN: The buckling of spherical shells by ex• ternal pressure, ,J. Aeron. Sci. i (19:!9), 4:!-;30; E. L. ltEISS, H. J. GREENB~:Ru, H. B. KELr.~:R: Nonlinear deflections of shallow spherical shells, ,J. Aeron. Sci. :!-1 (1957), 53:3-54:1. The stability of a hyperbolic p•uaboloid (Fig. 4.7) under its own weight ha,; been studied by A. R.H,STON: On the problem of buckling of a hyperbolic p•traboloi• dal shell loaded by its own weight, ,J. )[ath. Phys. :3;) (19511), 5:!-59.
Stanfortl Tht•st•s on Sht•ll 'l'ht•ory
The following list contains the Ph. D. theses and a few Engineer's theses whieh have been written under the author's direction at Stanford University. Extracts of some of them have appeared in print and hnve be:}n listed on the preceding pages. A few are still available as research reports, and microfilms of all of them are avail• able through the usual channels. The theses have been grouped by subjects and, in each group, are arranged chronologictlly. The year given is that of the degree award; some of the theses have been accepted in the second half of the preceding year. 1. Singular Solutions ot Shell Problems D. A. CONRAD (1957): Singular solutions in the theory of shallow shells (Ther• mal and force singularities, sphere, cylinder, hyperbolic paraboloid) K. J. FoR3BER':l (1961): Concentrated load on a shallow elliptic paraboloid (First solution for a shell of a rather general shape) W. R. BLYTHE (1962): Singular solutions in the theory of conical shells (Shallow cone, axisymmetric case of the steep cone) BIBLIOGRAPHY 51i
J. W. YouNG (196:3): Singular solutions for nonshallow dome shells (CompletE.' study of the stress singularities possible at the apex of a shell of revolution, proce• dure for patching a shallow-shell solution valid at the apex and an asymptotic solu• tion valid in the steeper part of the shell) R. 0. FoscHI (1966): Singular solutions in the theory of orthotropic plates (Cir• cular plates with an elastic anisotropy following the directions of a cartesian co• ordinate system, research undertaken to clarify the mathematical methods to be used in the two following theses) R. E. ELLING (1967): Singular solutions for shallow shells (Concentrated forces and hotspots in shallow paraboloids of positive or negative GAussian curvature). R. DORE ( 1969): Singular solutions for shallow shells (Paraboloids and cylinders, force singularities of the cylinder which are missing in CoNRAD's work) 2. Thermal Stt·esses D. A. CoNRAD: see above R. A. ErsENTRAUT (1958): Thermal stresses in cylindrical shells (Regular and singular solutions for partially heated cylinders, boundary value problem for a cylin· der with a hotspot, solutions for temperature distribution resulting from heat flow) G. E. STRICKLAND (1960): Temperature stresses in shells caused by local heat. ing (Differential equations of thermal stress in tensor form, including heat conduc• tion, solution for shells of constant GAussian curvature: cylinder, cone, sphere) 3. Paraboloids
~I. H. KASHANI-SAl!ET (1962): Membrane and bending theory of multi-span elliptic paraboloid shells (Structures consisting of two or more paraboloid shell>< connected by arch ribs) ~I. M. KATLA (Engineer's thesis, 1964): Membrane forces in structures consist• ing of elliptic and hyperbolic shells (Structure consisting of three shells, alter• natingly of positive and negative GAussian curvature, study of the admissible boundary conditions in a case where the field equations are elliptic in some domains and hyperbolic in others) R. VYAS ( 1966): Cut-outs in membrane shells (~Iembrane theory of the paraboloid of .Fig. 4.5 with a rectangular opening at the top, study of the stress singularity at the re-entrant corners) E. I. FmLD (1967): Membrane and bending theory of single and multi-span hyperbolic paraboloid shells (Shells as in Fig. 4.10, comparison of membrane and bending solutions, system of two shells with a connecting arch rib) 4. Cross Vaults E. G. DUARTE (Engineer's thesis, 1958): Membrane analysis of cross vaults (Tries to establish a theory similar to that of polygonal domes, shows the basie differences between these two types of composite shell stn1etures, singularities calling for further study) 0. Gii"REL (1961): Membrane analysis of cross vaults (Shows that slightly modified versions of DuARTE's cross vault have an acceptable membrane solution) }f. Toossr (1966): Bending analysis of cross vault shells (Establishes the compli• cated boundary conditions for the transition from one vault segment to the next, solution for a shell with a central opening) 5. Shells of Revolution R. E. PAULSEN (1953): Shells of negative curvature (Approximates shells of revolution of negative GAussian curvature by a sequence of cones, considers non• symmetric load, membrane and bending effects) 518 BIBLlOURAPHY
F. A. LEcKn; (W58): Bending theory for shells of revolution subjected to non• symmetric edge loads (Splits the 8th-order problem, derives one differential equa• tion for the oscillatory solutions and another one for the smooth solutions. Each equation is of the 4th order and can be used for digital computation) 0. C. DAVTDSON (1960): Nonsymmetric edge loads on a thin shell of negative curvature (Compares membrane and bending solutions for a one-sheet hyperboloid, explores the meaning of discontinuous membrane solutions) C. R. STEELE (1960): Toroidal shells with nonsymmetrio loading (Bending solu• tion describing the localized bending stresses at the top circle, far away from edge disturbances) R. F. HARTU)!G (1965): The deformation of orthotropic shells of revolution under nonsymmetrio edge loads (Principal rlirections of anisotropy along meri• dians and latitude circles, separate treatment of shallow and steep shells) ll. Various Pt•oblems F. T. GEYLING (W54): A general theory of deformations of membrane shells (Establishes the differential equation (4.45) and its boundary conditions, applies it to several examples) E. B. PAXSON (1963): Boundary value problems in the theory of shallow cylin• drical shells (Cylindrical shell with a circular hole) :\I. B. :\lARLOWE (1968): Some new developments in the foundations of shell theory (Tensor formulation of the basic equations of shell theory) T. FuKUSHIM:A ( 1969): Analysis of corrugated dome shells (Shallow shells of the form z = r"cosmO, singnlaritics at the apex being very different for m > tl an
A sphere 505 Adjacent equilibrium 4:33, 499 Buckling, cylinder Affine shells 179 axial compression 452 Almost cylindrical shell 490 external pressure 459 Almost plane circular plate :353 internal pressure 461 Almost spherical shell 370 nonuniform axial compression 4 78 Anisotropic shell 286, 295 shear 46:3 Arch 106 two-way compression 449 Asymptotic series 294 Buckling determinant 475 Asymptotic solution 334, 341, 388 Buckling diagram 451, 4ii:l, 4M, 457, Auxiliary variable 66 460,47U Axial r.ompression of a cylinder 452 Buckling load 436 see also Buckling, cylinder n Buckling, sphere Barrel vault Buckling mode 4:!6, 48:! bending theory 244, 265 Buckling, sphere 500 membrane theory 118 simplified bending theory 251 c Basic displacement, load, stress 4:l:l Cantilever shell 107 Beam analogue 119 CAUCHY-RlEMANN equations 54 Beam-column problem 484, 492 Centrifugal force :ltlO bei Change of curvature 211, :lli2 see THO)ISON functions Characteristics 72 Bellshaped shell 78 Circular cylinder Bending collapse 489 bending theory 204 BP-nding moment 6 membrane deformation 125 Bending rigidity 210,297,301, :!07, 310 membrane forces 110, 114, 118 Bending stiffness statically indeterminate shell127 see Bending rigidity Circular pipe 124 Bending theory 204 Circular plate 354, 358 ber Circular ring 507 see THO)ISON functions Coefficients for stress resultants BESSEL equation 291, :351, 377 cone 408 BESSEL function 291, 351, :378 cylinder 228, 242, 248, 252, 257 Boiler end sphere :1:18 ellipsoid 28 Colatitude 20 hemisphere :346 Collapse 434, 490 spherical cap 349 Columns, shell on- 49, 59, 70, 184, 231 Boundary conditions Combination of cylinder 107, 227, 449, 467 cylinder and cone 37, 380 hyperbolic paraboloid 175 cylinder and ellipsoid 35, 184 Buckling condition cylinder and sphere 37, 346, 349 cylinder 450, 4 72 sphere and cone 64 l~DEX 521
sphere and hyperboloid 78 Deflection of a pipe 124 Compatibility 161 Deformation Complex stress function 422 cylindrical shell 121, 206 Complex variable 54 folded plate structure 311 Compression test 485, 500 shallow shell 418 Concentrated couple 48, 56, 63 shell of arbitrary shape 197 Concentrated force shell of revolution 79, 319 cone 37,63 Deformity, shell with- :no shell of revolution 25 Deviation from exact shape 490 sphere 48, 53, 55, 356 Diaphragm 107, 124, 149, 162, 264 Conditions of equilibrium Differential equation(s) see Equilibrium conditions arbitrary shell 160 Conical roof 36 barrel vault 246 Conical shell buckling of cylinder 448, 498 bending theory 377, 38:3, 402 buckling of sphere 503 membrane theory 35, 61 conical shell :35, 61, 377, 405 Conical tank bottom :37, 380 cylindrical shell 105, 215, 217, 223, Convex shell 86 2:37 Cooling tower 231 cylindrical tank 271, 2\clll Corrugated pipe 2 elliptic equation Hi! Critical load hyperbolic equation 161 <~ee Buckling, cylinder membmne deformation 84, 91, 123, Buckling load 201 Buckling, sphere paraboli<' equation 161 Cross-grain modulus 2\J!i shallow paraboloid 42:1 Curvature shallow shell 421 see Change of curvature shell of revolution 22, 24, 41, 31i:3 Radius of curvature shell with ribs 310 Cut-out (hole) 355 sphere :127, :328, 342, :151, :387 Cycling loading 44:1 tank of variable wall thickness 290 Cylinder thin shell of revolution :169 anisotropic 295 water tank 271, 290 bending theory 204 Discontinuity stresses on columns 232 boiler end 346 membrane theory 103 hemisphere 345 with ring load 278, 282 Discrepancy of deformations 27, 3:1, with rings 284 121, 178 skew vault 196 Dished plate :153 stability 439 Displacement variable thickness 289 buckling 440 Cylindrical tank with cylinder 122, 206 clamped base 27:3 at edge 253 elastic roof 274 folded plate structure 312 elliptic bottom 184 shallow shell 418 elliptic croHs section 112 shell of revolution 82, :320 horizontal axis 112 Disturbed equilibrium 43:3 inclined axis 114 Dome on columns 49, 70, 2:H stiffening ring 288 Dome of constant strength 38 vertical axis 269 Dot see Dash D Double FoURJER series 222 Damped oscillations 294, 842, 393 Double-step formulas 24S Dash-and-dot notation 82, 204,317,403 Double-walled shell 298 INDEX
E Flexurnl rigidity Eccentricity 214, 425 Bee Bending rigidity Edge disturbance Folded plate structure 147, 311 cylinder 240, 274, 294, 488 Foot ring shell of revolution 70, 342 elliptic dome 191 Edge load polygonal dome 133 cylinder 116, 222, 236, 279 spherical dome 27, 51 hyperbolic paraboloid 174 FouRrER series hyperboloid of revolution 75 buckling 438, 466, 479, 491 shallow paraboloid 425 cylinder 115, 125, 222, 236 shell of revolution 46 double 222 sphere 332 folded plate structure 153, 311 Edge member shallow paraboloid 426 barrel vault 118 shell of revolution 42, 84 hyperboloid 169 sphere 388, 400 Edge shear 51, 151, 167, 174, 186 FOURIER sum 140 Effective shear force 230 Effective transverse force 2:30 G Effective width 287 GAussian curvature 71 Eigenvalue problem 434 Generator Elastic law cylinder 103 anisotropic shell 295 hyperbolic paraboloid 168, 17:3 cylinder 210, 212, 21:J hyperboloid of revolution 73 large deformation 497 Gridwork :302 shell of revolution 81, :J22, :J25, :361 sphere 326, 387 H Elastic modulus 81, 2911 Half ellipsoid 184 Ellipsoid of revolution 28, :J4, 183, 184 Half-filled pipe 258 Ellipsoid, triaxial 191 Hemispherical boiler end 346 Ellipsoidal shell 194 Hemispherical dome 49, 370, 376 Elliptic cylinder 112, 120 Hemispherical shell lOO Elliptic differential equation 161 Hexagonal dome 146 Elliptic dome 191 Hip 130 Elliptic foot ring 191 HooKE's law 122, 209 Elliptic paraboloid 164, 202 anisotropic 296 Elliptic parallels 189 with temperature terms 81 End disturbance 488 Hoop force 21, 105, 133 Energy method 433, 43!i Hoop strain 80, 122 Equilibrium conditions Hotspot arbitrary shell 1:j9 bending 4:31 cylinder 104, 205, 444 plane 429 shallow shell 416 Hydrostatic pressure 360 shell of revolution 20, 317, 324 Hyperbolic differential equation 161 sphere 326, 502 Hyperbolic paraboloid 168, 171 Eur,ER column 435, 4:37 Hyperboloid of revolution 71 Extensional deformation 91 Hypergeometric series 329 Extensional rigidity 81, 210, 297, 301, 307,:310 I External pressure 459 Imperfect cylinder 490 Imperfect sphere 370 F Improved membrane solution 236 Festoon curve 4ii3, 457, 460,476 Inclined cylinder 114 Fiber force 74 Incompatible deformation 33 INDEX 523
lnextensional deformation gridwork shell 302 cylinder 126 homogeneous shell 2 shell of revolution 84, 102 :\louR's circle 12 sphere 85, 392 Multipole 49 Infinite determinant472, 474, 477,481 ~Iushroom-shaped roof 36, 62 Internal pressure buckling of cylinder 461 N ellipsoid 192 Negative curvature 71, 168 elliptic cylinder 112 Neutral equilibrium 433,436 sphere 359 Nodal line 452,464 spherical boiler end 346 Nonconvex shell 86 toroid 31 Nonlinear theory of buckling 494 lNTZE tank 37 Nonregular polygonal dome 147 Isolated boundary 240, 247, 251 Nonuniform axial compression 478 Iterated coefficients 24!:1 Normal force 3 Normal point load 57 ,J Numerical integration 69 .Juncture of two shells :346 0 Oblique coordinates 1( 14 Octagonal dome 146 KELVIN functions Octagonal tube see THOMSON functions 154 Ogival dome Kinematic relations deformation 93 cylindrical shell 122, 209 membrane forces 29 nonlinear 495 One-sheet hyperboloid shallow shell 419 71 Oscillatory solutions 393 shell of arbitrary shape 198 shell of revolution 83, 321 p KmciiHOFF's force 230 Parabolic cylinder 427 Parabolic differential equation 161 [, Paraboloid of revolution 67, 8!:!, 161 Lantern ring 27, 130 Paraboloid shell 423 LAPLACE operator 217, 421 Particular solution Large deformation 494 cylindrical shell 218 Latitude circle 19 shell of revolution 359 LgnENDRE functions 504 Phase angle 280, 343 Line load 255, 256, 282, 2f!fi Piano hinge 311 Long cylinder 4 73 Pipe 127 Loss of solutions 42:!, 42!:! half-filled 258 octagonal 154 Jl Plate action 416 ~Iatrix 277 Plate strip, buckling of- 455, 476 ~!AXWELL's theorem 99, :382 Plywood shell 295 ~Iembrane force 8 Point load ~Iembrane theory 8 Bee Concentrated couple comparison with bending theory Concentrated force 392,425 Point moment ~IERCATOR's projection 54 see Concentrated couple )leridian 19 Pointed shell 29, 68 )feridional force 21 PmssoN's ratio 81 )!Iiddle surface Polygonal dome 129, 195 .anisotropic shell 299 Polygonal shell 163 524 INDEX
Potential energy 436 Shear force 4 Pressure vessel28, 112, 183, 190, 192, Shear load 256 :346, 349 Shear modulus 81 Prime-and-dot notation 82, 204, 317, Shear rigidity 297 403 Shear strain 81 Principal directions 11 Shell, definition 2 Principal forces 11 Shell operator 160, 421 Principal side 173 Shell of revolution 181 Principle of virtual displacements 436 Shell with ribs 308 Prismatic barrel vault 155 Short cylinder 475 Sign convention, MoHR's circle 13. R Simplified barrel vault theory 251 Radial line load Simply supported edge 227 on barrel vault 255 Singular solutions on finite cylinder 286 corre 37 on infinite cylinder 282 elliptic paraboloid 166 Radius of curvature 20 pointed shell 69 of ellipsoid 28 polygonal dome 146 Reciprocity of deformations 99, 382 thermal singularities 427 Reference vectors 44:1 toroid 31, 95 Regular load 130 see also Concentrated couple Regular polygonal dome 130 Concentrated force Reinforcing ring 284, 288 Hots pot Relaxation method 167 Sixth condition of equilibrium 21:1, 324,. Rib 299 416 Ridge beam 139, 170 Skew fiber force 14, 15, 74, 158 Rigid testing machine 500 Skew shearing force 1(; Rigid-body displacement 85, 487, 505 Skew vault 196 Rigidity Slightly dished circular plate :lii:3 sfe Bending rigidity Sludge digestion tank 380 Extensional rigidity Sphere Rigidity moment :301, 307, :Ho axisymmetric stresses 26, 326 Ring 106, 302, 507 buckling 500 see also Foot ring deformation 85 Lantern ring dome 26, 49 Stiffening ring edge load 49, 3:32, 344 Ring of radial forces 278, 282, 286 gas tank 59 Rounded apex 64 tank bottom :33, 340 thermal stresses :344 s unsymmetric stress system 4:3, :186· Secondary side 1n water tank 31, 344 Secondary stresses 214 Spherical :!:one 332 Self-equilibrating edge load 49 Splitting condition 368, 384 Semi-infinite cylinder 226, 279 Splitting of a differential equation Shallow cylinder 427 circular cylinder 291 Shallow paraboloid 422 shallow cylinder 428 Shallow shell 414 shallow paraboloid 42:3 Shallow sphere 353 shell of revolution :366, 368 Sharp edge 349 sphere 329, 393 Shear and axial compression 466 Square dome 146 Shear buckling 46:3 Stable equilibrium 433 Shear deformation in a cylinder 129 Statically indeterminate shell Shear edge 175 cylinder 127, 274 IXDEX 525
folded plate structure :H 1 Thermal expansion 81 pressure vessel 347 Thermal stress 344, 427 shell of revolution 100 see also Hotspot water tank 274 Thick shell 220, 3:33, 354 Stiffening ring 284, 288 Thin shell :321, 334, 354, 369 Strain THoMso~ functions 292, 351, 356, :ns, cylinder 122, 208 429 general shell 198 Toroidal shell :30, 94 shallow shell 419 Torsion of a cylinder 463 shell of revolution 80, :H9 Trajectories 11, 45, 52 Strain energy 95, 437 Transfer of edge loads 176 Stress discontinuity Transfer matrix 277 hyperbolic paraboloid 178 Transverse (shear) force 4 hyperboloid of revolution 75 Triangular shell 161 Stress function Tubular folded structure 154 AIRY'S 160,417 Twist 211 complex 422 Twisting moment 6, :J06 cylinder buckling 497 Twisting rigidity 297, :307, :310 Stress resultant 1, 2, 7, 209 Two-way compression 449 Stress singularities see Singular solutions u ::;tress trajectories Unit vector 443 see Trajectories Unstable equilibrium 434, 4:36 Stringer 299, :302, :308 Supporting ring :3:3 Surface of translation 166 V Variable thickness ·r cone 38:3 Tangential line load 256 cylinder 289 shell of revolution Tangential point load 48, 55 366, :n.t Variation of potential Tank energy 4:36 cylindrical 27:3 Vault action 416 on point supports 59 Vaulted hip roof 1:m spherical 32, 59 Virtual displacements 4:36 of variable thickness 289 Tank bottom w conical :37, :380 Water tank 31, 184, 269, 21:1, 21'\l elliptic 35, 184 Weight loading 41!9 spherical :3:3, 37, 340 Wind load 44