Skip to main content
SpringerLink
Log in

The Theory of Shells and Plates

  • Chapter
Linear Theories of Elasticity and Thermoelasticity

Abstract

A plate and more generally a shell is a special three-dimensional body whose boundary surface has special features. Although we defer defining a shell-like body in precise terms until Sect. 4, for the purpose of these preliminary remarks consider a surface—called a reference surface—and imagine material filaments from above and below surrounding the surface along the normal at each point of the reference surface. Suppose further that the bounding surfaces formed by the end points of the material filaments are equidistant from the reference surface. Such a three-dimensional body is called a shell if the dimension of the body along the normals, called the thickness, is small. A shell is said to be thin if its thickness is much smaller than a certain characteristic length of the reference surface, e.g., the minimum radius of the curvature of the reference surface for initially curved shells.2

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Kirchhoff, G.: 1850 Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. Crelles J. 40, 51–88 = Ges. Abh. 237-279. (1, 7, 20, 26).

    MATH  Google Scholar 

  2. 1886 Todhunter, I., and K. Pearson: A history of the theory of elasticity, Vol. I. Cambridge: University Press = reprint, Dover Publications. (1, 20).

    Google Scholar 

  3. Love, A. E. H.: 1888 The small free vibrations and deformation of a thin elastic shell. Phil. Trans. Roy. Soc. London, Ser. A 179, 491–546. (1, 4, 7, 12 A, 16, 21, 21 A).

    MATH  Google Scholar 

  4. Rayleigh, (Lord): 1888 On the bending and vibration of thin elastic shells, especially of cylindrical form. Proc. Roy. Soc. London, Ser. A 45, 105–123. (21 A).

    MATH  Google Scholar 

  5. Basset, A.B.: 1890 On the extension and flexure of cylindrical and spherical thin elastic shells. Phil. Trans. Roy. Soc. London, Ser. A 181, 433–480. (12A, 21 A).

    MATH  Google Scholar 

  6. Lamb, H.: 1890 On the deformation of an elastic shell. Proc. London Math. Soc. 21, 119–146. (12A, 21A).

    MATH  MathSciNet  Google Scholar 

  7. Love, A. E. H.: 1891 Note on the present state of the theory of thin elastic shells. Proc. Roy. Soc. London, Ser. A 49, 100–102. (21 A).

    MATH  Google Scholar 

  8. 1892 Love, A. E. H.: A treatise on the mathematical theory of elasticity 1. Cambridge. ix+354pp. (1).

    Google Scholar 

  9. Duhem, P.: 1893 Le potentiel thermodynamique et la pression hydrostatique. Ann. Ecole Norm. (3) 10, 187–230. (1, 4).

    MathSciNet  Google Scholar 

  10. 1893 Love, A. E. H.: A treatise on the mathematical theory of elasticity 2. Cambridge. xi+327 pp. (1, 12A, 21A).

    Google Scholar 

  11. 1893 Todhunter, L, and K. Pearson: A history of the theory of elasticity, Vol. II (Parts I and II). Cambridge: University Press = reprint, Dover Publications. (1, 20).

    Google Scholar 

  12. FÖppl, A.: 1907 Vorlesungen über Technische Mechanik, Vol. 5. München: R. Oldenbourg. (21).

    MATH  Google Scholar 

  13. Cosserat, E., et F.: 1908 Sur la théorie des corps minces. Compt. Rend. 146, 169–172. (1).

    MATH  Google Scholar 

  14. 1909 Cosserat, E., et F.: Théorie des Corps Déformables. Paris, vi-f 226 pp. = Appendix, pp. 953-1173, of Chwolson’s Traité de Physique, 2nd ed. Paris. (1, 4).

    Google Scholar 

  15. Karman, Th. v.: 1910 Festigkeitsprobleme im Maschinenbau. Encyklopädie der Mathematischen Wissenschaften, Vol. 4/4, pp. 311-385 = Collected Works, 1, 141–207. (21).

    Google Scholar 

  16. Hellinger, E.: 1914 Die allgemeinen Ansätze der Mechanik der Kontinuaa. Encyklopädie der Mathematischen Wissenschaften, Vol. 4/4, pp. 601–694. (7).

    Google Scholar 

  17. Lamb, H.: 1917 On waves in an elastic plate. Proc. Roy. Soc. London, Ser. A 93, 114–128. (20, 24).

    MATH  Google Scholar 

  18. Nadai, A.: 1925 Die elastischen Platten. Berlin: Springer. viii + 326pp. (20).

    MATH  Google Scholar 

  19. 1927 Love, A. E. H.: 4th ed. of [1892, 1] and [1893, 2]. (1).

    Google Scholar 

  20. McConnell, A. J.: 1931 Applications of the absolute differential calculus. London and Glasgow: Blackie & Son. xii + 318pp. (Appendix, A.2, A.4).

    MATH  Google Scholar 

  21. Flügge, W.: 1932 Die Stabilität der Kreiszylinderschale. Ing.-Arch. 3, 463–506. (21, 21A).

    MATH  Google Scholar 

  22. 1933 Donnell, L. H.: Stability of thin-walled tubes under torsion. NACA Tech. Rept. No. 479. (21).

    Google Scholar 

  23. Flügge, W.: 1934 Statik und Dynamik der Schalent. Berlin: Springer. vii + 240pp. (21A).

    MATH  Google Scholar 

  24. Marguerre, K.: 1935 Thermo-elastische Plattengleichungen. Z. Angew. Math. Mech. 15, 369–372. (18).

    MATH  Google Scholar 

  25. Trefftz, E.: 1935 Ableitung der Schalenbiegungsgleichungen mit dem Castiglianoschen Prinzip. Z. Angew. Math. Mech. 15, 101–108. (21).

    MATH  Google Scholar 

  26. Goodier, J. N.: 1936 The influence of circular and elliptical holes on the transverse flexure of elastic plates. Phil. Mag., Ser. VII 22, 69–80. (20).

    MATH  Google Scholar 

  27. Odqvist, F.: 1937 Equations complètes de l’équilibre des couches minces élastiques gauches. Compt. Rend. 205, 271–273. (7).

    MATH  Google Scholar 

  28. Goodier, J. N.: 1938 On the problems of the beam and the plate in the theory of elasticity. Trans. Roy. Soc. Can., Ser. III 32, 65–88. (21).

    Google Scholar 

  29. 1938 Marguerre, K.: Zur Theorie der gekrümmten Platte großer Formänderung. Proc. Fifth Intern. Congr. Appl. Mech. (Cambridge, Mass.), pp. 93-101. (21).

    Google Scholar 

  30. Gol’denveizer, A. L.: 1940 Equations of the theory of thin shells [in Russian]. Prikl. Mat. Mekh. 4, 35–42. (7, 26).

    Google Scholar 

  31. Lur’e, A. I.: 1940 General theory of thin elastic shells [in Russian]. Prikl. Mat. Mekh. 4, 7–34. (21 A, 26).

    Google Scholar 

  32. Reissner, E.: 1941 A new derivation of the equations for the deformation of elastic shells. Amer. J. Math. 63, 177–184. (12A, 21A).

    MathSciNet  Google Scholar 

  33. 1941 Synge, J. L., and W. Z. Chien: The intrinsic theory of elastic shells and plates. Th. von Kármán Anniv. Vol., pp. 103-120. (12, 12A).

    Google Scholar 

  34. Reissner, E.: 1942 Note on the expressions for the strains in a bent thin shell. Amer. J. Math. 64, 768–772. (21 A).

    MATH  MathSciNet  Google Scholar 

  35. Novozhilov, V.: 1943 On an error in a hypothesis of the theory of shells. Compt. Rend. Acad. Sci. SSSR (Doklady) (N.S.) 38, 160–164. (12A).

    Google Scholar 

  36. Novozhilov V., and R. Finkel’shtein: 1943 On the incorrectness of Kirchhoff’s hypotheses in the theory of shells [in Russian]. Prikl. Mat. Mekh. 7, 331–340. (12A).

    Google Scholar 

  37. Byrne, R.: 1944 Theory of small deformations of a thin elastic shell. Seminar Reports in Mathematics. Univ. of Calif. (Los Angeles) Publ. in Math. (N.S.) 2, 103–152. (21 A, 26).

    MathSciNet  Google Scholar 

  38. Chien, W. Z.: 1944 The intrinsic theory of thin shells and plates (Parts I, II and III). Quart. Appl. Math. 1, 297–327; 2, 43-59 and 120-135. (12, 21A).

    MathSciNet  Google Scholar 

  39. Gol’denveizer, A. L.: 1944 On the applicability of general theorems of the theory of elasticity to thin shells [in Russian]. Prikl. Mat. Mekh. 8, 3–14. (26).

    Google Scholar 

  40. Love, A. E. H.: 1944 A treatise on the mathematical theory of elasticity (reprint of [1927, 1]). New York: Dover Publications. xviii + 643pp. (1, 12A, 20, 21 A, 26).

    Google Scholar 

  41. Reissner, E.: 1944 On the theory of bending of elastic plates. J. Math. Phys. 23, 184–191. (20).

    MATH  MathSciNet  Google Scholar 

  42. Vlasov, V. Z.: 1944 Basic differential equations in general theory of elastic shells [in Russian]. Prikl. Mat. Mekh. 8, 109–140. [English Transi.: NACA TM 1241 (1951) 58 pp.] (21 A).

    MATH  MathSciNet  Google Scholar 

  43. Bromberg, E., and J. J. Stoker: 1945 Non-linear theory of curved elastic sheets. Quart. Appl. Math. 3, 246–265. (21).

    MATH  MathSciNet  Google Scholar 

  44. Reissner, E.: 1945 The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12, A–69–77. (20, 24).

    MathSciNet  Google Scholar 

  45. Truesdell, C.: 1945 The membrane theory of shells of revolution. Trans. Am. Math. Soc. 58, 96–166. (12A, 21A).

    MATH  MathSciNet  Google Scholar 

  46. 1946 Drucker, D. C.: Discussion of [1945, 2]. J. Appl. Mech. 13, A-250. (20).

    Google Scholar 

  47. 1946 Goodier, J. N.: Discussion of [1945, 2]. J. Appl. Mech. 13, A-251. (20).

    Google Scholar 

  48. 1946 NovozHiLov, V. V.: A new method for the analysis of thin shells [in Russian]. Izv. Akad. Nauk SSR Otd. Tekhn. Nauk, No. 1. (20, 21 A).

    Google Scholar 

  49. Bolle, L.: 1947 Contribution au problème linéaire de flexion d’une plaque élastique (Parts 1 and 2). Bull. Tech. Suisse Romande 73, 281–285 and 293-298. (20).

    MathSciNet  Google Scholar 

  50. 1947 Eisenhart, L. P.: An introduction to differential geometry. Princeton Univ. Press. x+ 304 pp. (Appendix, A.2).

    Google Scholar 

  51. Gol’denveizer, A. L., and A. I. Lur’e: 1947 On the mathematical theory of the equilibrium of elastic shells [in Russian]. (Review of works published in the U.S.S.R.). Prikl. Mat. Mekh. 11, 565–592. (7, 21 A, 26).

    MathSciNet  Google Scholar 

  52. 1947 Lur’e, A. I.: Statics of thin-walled elastic shells [transi, from 1947 Russian ed.]. AEC-tr-3798 (U.S. Atomic Energy Commission, Tech. Info. Service). iv + 210pp. (21A).

    Google Scholar 

  53. Reissner, E.: 1947 On bending of elastic plates. Quart. Appl. Math. 5, 55–68. (20).

    MATH  MathSciNet  Google Scholar 

  54. Uflyand, Ia. S.: 1948 Wave propagation with transverse vibrations of bars and plates [in Russian]. Prikl. Mat. Mekh. 12, 287–300. (20).

    MathSciNet  Google Scholar 

  55. 1949 Friedrichs, K. O.: The edge effect in the bending of plates. H. Reissner Anniversary Volume, pp. 197-210. Ann Arbor, Mich.: J. W. Edwards. (21).

    Google Scholar 

  56. Green, A. E.: 1949 On Reissner’s theory of bending of elastic plates. Quart. Appl. Math. 7, 223–228. (20).

    MATH  MathSciNet  Google Scholar 

  57. Green, A. E.: 1949 The elastic equilibrium of isotropic plates and cylinders. Proc. Roy. Soc. London, Ser. A 195, 533–552. (20).

    MATH  MathSciNet  Google Scholar 

  58. 1949 Hildebrand, F. B., E. Reissner, and G. B. Thomas: Notes on the foundations of the theory of small displacements of orthotropic shells. NACA TN 1833. (7, 20, 21A).

    Google Scholar 

  59. 1949 Synge, J. L., and A. Schild: Tensor calculus. Toronto: Univ. of Toronto Press, xi + 324 pp. (Appendix, A.1, A.3).

    Google Scholar 

  60. Zerna, W.: 1949 Beitrag zur allgemeinen Schalenbiegetheorie. Ing.-Arch. 17, 149–164. (12A).

    MATH  MathSciNet  Google Scholar 

  61. Friedrichs, K. O.: 1950 Kirchhoff’s boundary conditions and the edge effect for elastic plates. Proc. Symp. Appl. Math. 3, 117–124. (21).

    Google Scholar 

  62. Green, A. E., and W. Zerna: 1950 The equilibrium of thin elastic shells. Quart. J. Mech. Appl. Math. 3, 9–22. (20, 21 A).

    MATH  MathSciNet  Google Scholar 

  63. Lur’e, A. I.: 1950 On the equations of the general theory of elastic shells [in Russian]. Prikl. Mat. Mekh. 14, 558–560. (26).

    MathSciNet  Google Scholar 

  64. Parkus, H.: 1950 Die Grundgleichungen der Schalentheorie in allgemeinen Koordinaten. Oesterr. Ing.-Arch. 4, 160–174 and 6 (1952), 72. (7).

    MATH  MathSciNet  Google Scholar 

  65. Reissner, E.: 1950 On a variational theorem in elasticity. J. Math. Phys. 29, 90–95. (7, 20).

    MATH  MathSciNet  Google Scholar 

  66. Reissner, E.: 1950 On axisymmetrical deformations of thin shells of revolution. Proc. Symp. Appl. Math. 3, 27–52. (21).

    MathSciNet  Google Scholar 

  67. Mindlin, R. D.: 1951 Influence of rotatory inertia and shear on flexural vibrations of isotropic, elastic plates. J. Appl. Mech. 18, 31–38. (20, 24).

    MATH  Google Scholar 

  68. Mindlin, R. D.: 1951 Thickness-shear and flexural vibrations of crystal plates. J. Appl. Phys. 22, 316–323. (20).

    MATH  MathSciNet  Google Scholar 

  69. 1951 Timoshenko, S., and J. N. Goodier: Theory of elasticity, 2nd ed. McGraw-Hill, xviii + 506 pp. (24).

    Google Scholar 

  70. 1952 Eringen, A.C.: On the non-linear vibration of circular membrane. Proc. First U.S. Nat’l. Congr. Appl. Mech. (Chicago, 1951), pp. 139-145. (21).

    Google Scholar 

  71. Reissner, E.: 1952 Stress-strain relations in the theory of thin elastic shells. J. Math. Phys. 31, 109–119. (20, 21A).

    MATH  MathSciNet  Google Scholar 

  72. Schäfer, M.: 1952 Über eine Verfeinerung der klassischen Theorie dünner schwach gebogener Platten. Z. Angew. Math. Mech. 32, 161–171. (20).

    MATH  MathSciNet  Google Scholar 

  73. Melan, E., and H. Parkus: 1953 Wärmespannungen infolge stationärer Temperaturfelder. Wien: Springer. v+114pp. (18).

    Google Scholar 

  74. 1953 Naghdi, P. M., and J. C. Rowley: On the bending of axially symmetric plates on elastic foundations. Proc. First Midwestern Conf. on Solid Mechanics (Univ. of Illinois), pp. 119-123. (20).

    Google Scholar 

  75. Novozhilov, V. V.: 1953 Foundations of the nonlinear theory of elasticity [transi, from the 1947 Russian ed.]. Rochester, N.Y.: Graylock Press, vi + 233 pp. (21).

    Google Scholar 

  76. Reissner, E.: 1953 On a variational theorem for finite elastic deformations. J. Math. Phys. 32, 129–135. (7, 21).

    MATH  MathSciNet  Google Scholar 

  77. Truesdell, C.: 1953 Review of [1953, 3]. Bull. Am. Math. Soc. 59, 467–473. (21, 21 A).

    MathSciNet  Google Scholar 

  78. Truesdell, C.: 1953 The physical components of vectors and tensors. Z. Angew. Math. Mech. 33, 345–356. (A.4).

    MATH  MathSciNet  Google Scholar 

  79. Green, A. E., and W. Zerna: 1954 Theoretical elasticity. Oxford: Clarendon Press. xiii+442 pp. (4, 12A).

    MATH  Google Scholar 

  80. Truesdell, C.: 1954 Remarks on the paper “The physical components of vectors and tensors.” Z. Angew. Math. Mech. 34, 69–70. (A.4).

    MATH  MathSciNet  Google Scholar 

  81. Eringen, A. C.: 1955 On the nonlinear oscillations of viscoelastic plates. J. Appl. Mech. 22, 563–567. (21).

    MATH  Google Scholar 

  82. Fung, Y. C, and W. H. Witrick: 1954 A boundary layer phenomenon in the large deflexion of thin plates. Quart. J. Mech. Appl. Math. 8, 191–210. (21).

    Google Scholar 

  83. Hoff, N. J.: 1954 The accuracy of Donnell’s equations. J. Appl. Mech. 22, 329–334. (22).

    MathSciNet  Google Scholar 

  84. Hu, H.-C.: 1954 On some variational principles in the theory of elasticity and plasticity. Scientia Sinica 4, 33–54. (7).

    Google Scholar 

  85. Mansfield, E. H.: 1954 The inextensional theory for thin flat plates. Quart. J. Mech. Appl. Math. 8, 338–352. (21).

    MathSciNet  Google Scholar 

  86. 1954 Mindlin, R. D.: An introduction to the mathematical theory of vibration of elastic plates. U. S. Army Signal Corps Engineering Laboratories Monograph, Ft. Mon-mouth, N.J. (20).

    Google Scholar 

  87. 1954 Washizu, K.: On the variational principles of elasticity and plasticity. Tech. Rept. 25-18, Contract N 50 ri-07833, Mass. Inst. of Tech. (7).

    Google Scholar 

  88. Frederick, D.: 1956 On some problems in bending of thick circular plates on an elastic foundation. J. Appl. Mech. 23, 195–200. (20).

    MATH  Google Scholar 

  89. 1956 Sokolnikoff, I. S.: Mathematical theory of elasticity, 2nd ed. McGraw-Hill. xi + 476 pp. (26).

    Google Scholar 

  90. Alblas, J. B.: 1957 Theorie van De Driedimensionale Spanningstoestand in een Doorboorde Plaat. Amsterdam: H. J. Paris. viii+ 127 pp. (20).

    Google Scholar 

  91. Ashwell, D. G.: 1957 The equilibrium equations of the inextensional theory for thin flat plates. Quart. J. Mech. Appl. Math. 10, 169–182. (21).

    MATH  MathSciNet  Google Scholar 

  92. 1957 Mindlin, R. D., and M. Onoe: Mathematical theory of vibrations of elastic plates. Proc. 11th Ann. Symp. on Frequency Control (U.S. Army Signal Engineering Laboratories, Fort Monmouth, N.J.), pp. 17-40. (20).

    Google Scholar 

  93. Naghdi, P. M.: 1957 On the theory of thin elastic shells. Quart. Appl. Math. 14, 369–380. (20, 21 A).

    MATH  MathSciNet  Google Scholar 

  94. Ericksen, J. L., and C. Truesdell: 1958 Exact theory of stress and strain in rods and shells. Arch. Rational Mech. Anal. 1, 295–323. (4, 5, 9, 12A).

    MATH  MathSciNet  Google Scholar 

  95. 1958 Essenburg, F., and P. M. Naghdi: On elastic plates of variable thickness. Proc. 3rd U.S. Nat’l. Congr. Appl. Mech. (Providence, R.I.), pp. 313-319. (20, 24).

    Google Scholar 

  96. Knowles, J. K., and E. Reissner: 1958 Notes on stress-strain relations for thin elastic shells. J. Math. Phys. 37, 269–282. (21, 21 A).

    MATH  MathSciNet  Google Scholar 

  97. Noll, W.: 1958 A mathematical theory of the mechanical behavior of continuous media. Arch. Rational Mech. Anal. 2, 197–226. (13).

    MATH  Google Scholar 

  98. Vlasov, V. Z.: 1958 Allgemeine Schalentheorie und ihre Anwendung in der Technik [transi, from the 1949 Russian ed.]. Berlin: Akademie-Verlag, xi + 661 pp. (21, 21 A).

    Google Scholar 

  99. Johnson, M. W., and E. Reissner: 1959 On the foundations of the theory of thin elastic shells. J. Math. Phys. 37, 371–392. (21).

    MATH  MathSciNet  Google Scholar 

  100. Morgenstern, D.: 1959 Herleitung der Plattentheorie aus der dreidimensionalen Elastizitätstheorie. Arch. Rational Mech. Anal. 4, 411–417. (20).

    MathSciNet  Google Scholar 

  101. Novozhilov, V. V.: 1959 The theory of thin shells [transi, from the 1951 Russian ed.]. Groningen: P. Noordhoof Ltd. vi + 376 pp. (7, 21, 21 A, 26).

    Google Scholar 

  102. Rivlin, R. S.: 1959 The deformation of a membrane formed by inextensible cords. Arch. Rational Mech. Anal. 2, 447–476. (14).

    MathSciNet  Google Scholar 

  103. Rüdiger, D.: 1959 Zur Theorie elastischer Schalen. Ing.-Arch. 28, 281–288. (7).

    MATH  MathSciNet  Google Scholar 

  104. 1959 Sanders, J. L. Jr.: An improved first-approximation theory for thin shells. NASA Tech. Rept. R-24. 11 pp. (7, 20, 26).

    Google Scholar 

  105. Timoshenko, S., and W. Woinowsky-Kreiger: 1959 Theory of plates and shells, 2nd ed. New York: McGraw-Hill. xiv+ 580 pp. (20, 21).

    Google Scholar 

  106. Willmore, T. J.: 1959 An introduction to differential geometry. Oxford: Clarendon Press, x + 317 pp. (Appendix, A.2).

    MATH  Google Scholar 

  107. Bolotin, V. V.: 1960 Equation for the non-stationary temperature fields in thin shells in the presence of sources of heat. J. Appl. Math. Mech. (Transi, of PMM) 24, 515–519. (18).

    MATH  MathSciNet  Google Scholar 

  108. 1960 Cohen, J. W.: The inadequacy of the classical stress-strain relations for the right hélicoïdal shell. Proc. IUTAM Symposium on the theory of thin elastic shells (Delft, 1959), pp. 415-433. (21).

    Google Scholar 

  109. Ericksen, J. L.: 1960 Appendix. Tensor fields. Handbuch der Physik, Vol. III/l (edit, by S. Flügge), pp. 794–858. Berlin-Göttingen-Heidelberg: Springer. (Appendix, A.4).

    Google Scholar 

  110. Flügge, W.: 1960 Stresses in shells. Berlin-Göttingen-Heidelberg: Springer. xi + 499pp. (21A).

    MATH  Google Scholar 

  111. Green, A. E., and J. E. Adkins: 1960 Large elastic deformations. Oxford: Clarendon Press. vii + 348pp. (14, 17, 21).

    MATH  Google Scholar 

  112. 1960 Koiter, W. T.: A consistent first approximation in the general theory of thin elastic shells. Proc. IUTAM Symposium on the theory of thin elastic shells (Delft, 1959), PP. 12-33. (20, 21).

    Google Scholar 

  113. Mikolwitz, J.: 1960 Flexural stress waves in an infinite elastic plate due to suddenly applied concentrated transverse load. J. Appl. Mech. 27, 681–689. (20).

    MathSciNet  Google Scholar 

  114. 1960 Mindlin, R. D.: Waves and vibrations in isotropic, elastic plates. Proc. First Symp. on Naval Structural Mechanics (Stanford, Calif., 1958), p. 199-232. (20).

    Google Scholar 

  115. Naghdi, P. M.: 1960 A note on rigid body displacements in the theory of thin elastic shells. Quart. Appl. Math. 18, 296–298. (21 A).

    MATH  MathSciNet  Google Scholar 

  116. Naghdi, P. M.: On Saint Venant’s principle: 1960 elastic shells and plates. J. Appl. Mech. 27, 417–422. (21 A, 26).

    MATH  MathSciNet  Google Scholar 

  117. Oniashvili, O. D.: 1960 Analysis of shells and other thin-walled spatial structures. Chap. 5 of Structural mechanics in the U.S.S.R., 1917-1957, pp. 223–275. Oxford: Pergamon Press. (21).

    Google Scholar 

  118. Reiss, E. L.: 1960 A theory for the small rotationally symmetric deformations of cylindrical shells. Comm. Pure Appl. Math. 13, 531–550. (21).

    MATH  MathSciNet  Google Scholar 

  119. 1960 Reissner, E.: On some problems in shell theory. Proc. First Symp. on Naval Structural Mechanics (Stanford, Calif., 1958), pp. 74-114. (21).

    Google Scholar 

  120. 1960 Truesdell, C., and R. Toupin: The classical field theories. Handbuch der Physik, Vol. III/l (edit, by S. Flügge), pp. 226-793-Berlin-Göttingen-Heidelberg: Springer. (2, 4, 8, 9, 12A, 13, 21, 26).

    Google Scholar 

  121. 1960 Zerna, W.: Über eine nichtlineare allgemeine Theorie der Schalen. Proc. IUTAM Symposium on the theory of thin elastic shells (Delft, 1959), PP. 34-42. (21).

    Google Scholar 

  122. Ericksen, J.L.: 1961 Conservation laws for liquid crystals. Trans. Soc. Rheol. 5, 23–34. (4, 8).

    MATH  MathSciNet  Google Scholar 

  123. Friedrichs, K. O., and R. F. Dressler: 1961 A boundary-layer theory for elastic plates. Comm. Pure Appl. Math. 14, 1–33. (21).

    MATH  MathSciNet  Google Scholar 

  124. Gol’denveizer, A. L.: 1961 Theory of elastic thin shells [transi, from the 1953 Russian ed.]. Oxford: Pergamon Press, xxi + 658 pp. (7, 21, 21 A, 26).

    Google Scholar 

  125. Günther, W.: 1961 Analoge Systeme von Schalengleichungen. Ing.-Arch. 30, 160–186. (6).

    MATH  MathSciNet  Google Scholar 

  126. Koiter, W. T.: 1961 A systematic simplification of the general equations in the linear theory of thin shells. Proc. Koninkl. Ned. Akad. Wetenschap., Ser. B 64, 612–619. (20).

    MATH  Google Scholar 

  127. 1961 Leonard, R. W.: Nonlinear first approximation thin shell and membrane theory. (National Aeronautics and Space Administration, Langley Field, Va.) (21).

    Google Scholar 

  128. 1961 Lur’e, A. I.: On the static geometric analogue of shell theory. Problems of continuum mechanics (N. I. Muskhelishvili Anniv. Vol.), pp. 267-274. Phila.: S.I.A.M. (26).

    Google Scholar 

  129. Mindlin, R. D.: 1961 High frequency vibrations of crystal plates. Quart. Appl. Math. 19, 51–61. (20).

    MATH  MathSciNet  Google Scholar 

  130. Morgenstern, D., and I. Szabô: 1961 Vorlesungen über theoretische Mechanik. Berlin-Göttingen-Heidelberg: Springer. (20).

    MATH  Google Scholar 

  131. 1961 Mushtari, Kh. M., and K. Z. Galimov: Non-linear theory of thin elastic shells [transi, from the 1957 Russian ed.]. Israel Program for Scientific Translations. 374 pp. (21).

    Google Scholar 

  132. Reiss, E. L., and S. Locke: 1961 On the theory of plane stress. Quart. Appl. Math. 19, 195–203. (21).

    MATH  MathSciNet  Google Scholar 

  133. Rüdiger, D.: 1961 Eine geometrisch nichtlineare Schalentheorie. Z. Angew. Math. Mech. 41, 198–207. (21).

    MathSciNet  Google Scholar 

  134. Duddeck, H.: 1962 Das Randstörungsproblem der technischen Biegetheorie dünner Schalen in drei korrespondierenden Darstellungen. Oester. Ing.-Arch. 17, 32–57. (7).

    MATH  MathSciNet  Google Scholar 

  135. Essenburg, F.: 1962 On surface constraints in plate problems. J. Appl. Mech. 29, 340–344. (20).

    MATH  Google Scholar 

  136. Gol’denveizer, A. L.: 1962 Derivation of an approximate theory of bending of a plate by the method of asymptotic integration of the equations of the theory of elasticity. J. Appl. Math. Mech. (Transi, of PMM) 26, 1000–1025. (21).

    MathSciNet  Google Scholar 

  137. Green, A. E.: 1962 On the linear theory of thin elastic shells. Proc. Roy. Soc. London, Ser. A 266, 143–160. (21).

    MATH  Google Scholar 

  138. Green, A. E.: 1962 Boundary-layer equations in the theory of thin elastic shells. Proc. Roy. Soc. London, Ser. A 269, 481–491. (21).

    Google Scholar 

  139. 1962 Nordgren, R. P., and P. M. Naghdi: Propagation of thermoelastic waves in an unlimited shallow spherical shell under heating. Proc. 4th U. S. Nat’l. Congr. Appl. Mech. (Berkeley, Calif.), pp. 311-324. (18).

    Google Scholar 

  140. Reiss, E. L.: 1962 On the theory of cylindrical shells. Quart. J. Mech. Appl. Math. 15, 324–338. (21).

    MathSciNet  Google Scholar 

  141. Zerna, W.: 1962 Mathematisch strenge Theorie elastischer Schalen. Z. Angew. Math. Mech. 42, 333–341. (7).

    MATH  MathSciNet  Google Scholar 

  142. 1963 Budiansky, B., and J. L. Sanders, Jr.: On the “best” first-order linear shell theory. Progress in applied mechanics (The Prager Anniv. Vol.), pp. 129-140. (20).

    Google Scholar 

  143. Coleman, B. D., and W. Noll: 1963 The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rational Mech. Anal. 13, 167–178. (13).

    MATH  MathSciNet  Google Scholar 

  144. Gol’denveizer, A. L.: 1963 Derivation of an approximate theory of shells by means of asymptotic integration of the equations of the theory of elasticity. J. Appl. Math. Mech. (Transi, of PMM) 27, 903–924. (21).

    MathSciNet  Google Scholar 

  145. Johnson, M. W.: 1963 A boundary layer theory of unsymmetric deformations of circular cylindrical elastic shells. J. Math. Phys. 42, 167–188. (21).

    MATH  Google Scholar 

  146. 1963 Mindlin, R. D.: High frequency vibrations of plated, crystal plates. Progress in applied mechanics (The Prager Anniv. Vol.), pp. 73-84. (20).

    Google Scholar 

  147. Naghdi, P. M.: 1963 Foundations of elastic shell theory. Progress in solid mech., Vol. 4 (edit, by I.N. Sneddon and R. Hill), pp. 1–90. Amsterdam: North-Holland Publ. Co. =Tech. Rept. No. 15, Contract Nonr-222 (69), Univ. of Calif., Berkeley (Jan. 1962). (7, 12, 12A, 20, 21 A, 26, A.3).

    Google Scholar 

  148. Naghdi, P. M.: 1963 A new derivation of the general equations of elastic shells. Int. J. Engng. Sci. 1, 509–522. (6, 7, 15, 20, 26).

    MathSciNet  Google Scholar 

  149. Naghdi, P. M., and R. P. Nordgren: 1963 On the nonlinear theory of elastic shells under the Kirchhoff hypothesis. Quart. Appl. Math. 21, 49–59. (15, 21).

    MATH  MathSciNet  Google Scholar 

  150. Reissner, E.: 1963 On the derivation of boundary conditions for plate theory. Proc. Roy. Soc. London, Ser. A 276, 178–186. (21).

    MATH  MathSciNet  Google Scholar 

  151. Reissner, E.: 1963 On the derivation of the theory of thin elastic shells. J. Math. Phys. 42, 263–277. (21).

    Google Scholar 

  152. 1963 Reissner, E.: On the equations for finite symmetrical deflections of thin shells of revolution. Progress in applied mechanics (The Prager Anniv. Vol.), pp. 171-178. (21).

    Google Scholar 

  153. Sanders, J. L., Jr.: 1963 Nonlinear theories for thin shells. Quart. Appl. Math. 21, 21–36. (21).

    MathSciNet  Google Scholar 

  154. Serbin, H.: 1963 Quadratic invariants of surface deformations and the strain energy of thin elastic shells. J. Math. Phys. 4, 838–851. (10).

    MathSciNet  Google Scholar 

  155. 1963 Stoker, J. J.: Elastic deformations of thin cylindrical sheets. Progress in applied mechanics (The Prager Anniv. Vol.), pp. 179-188. (21).

    Google Scholar 

  156. Tiffen, R., and P. G. Lowe: 1963 An exact theory of generally loaded elastic plates in terms of moments of the fundamental equations. Proc. London Math. Soc. 13, 653–671. (12).

    MATH  MathSciNet  Google Scholar 

  157. Wainwright, W. L.: 1963 On a nonlinear theory of elastic shells. Int. J. Engng. Sci. 1, 339–358. (21).

    MathSciNet  Google Scholar 

  158. 1964 Ambartsumian, S. A.: Theory of anisotropic shells [transi, from a 196I Russian ed.]. NASA Technical Translation TTF-118. vi + 395 pp. (21).

    Google Scholar 

  159. Fox, N.: 1964 On asymptotic expansions in plate theory. Proc. Roy. Soc. London, Ser. A 278, 228–233. (21).

    MATH  Google Scholar 

  160. Green, A. E., and R. S. Rivlin: 1964 On Cauchy’s equations of motion. Z. Angew. Math. Phys. 15, 290–292. (8).

    MATH  MathSciNet  Google Scholar 

  161. 1964 Naghdi, P. M.: On the nonlinear thermoelastic theory of shells. Proc. IASS Symposium on non-classical shell problems (Warsaw, 1963), pp. 5-26. (11).

    Google Scholar 

  162. Naghdi, P. M.: 1964 Further results in the derivation of the general equations of elastic shells. Int. J. Engng. Sci. 2, 269–273. (20, 26).

    MathSciNet  Google Scholar 

  163. Naghdi, P. M.: 1964 On a variational theorem in elasticity and its application to shell theory. J. Appl. Mech. 31, 647–653. (7, 20).

    MATH  MathSciNet  Google Scholar 

  164. Reissner, E.: 1964 On the form of variationally derived shell equations. J. Appl. Mech. 31, 233–238. (7, 20).

    MATH  MathSciNet  Google Scholar 

  165. Toupin, R. A.: 1964 Theories of elasticity with couple-stress. Arch. Rational Mech. Anal. 17, 85–112. (4).

    MATH  MathSciNet  Google Scholar 

  166. Gol’denveizer, A., and A. V. Kolos: 1965 On the derivation of two-dimensional equations in the theory of thin elastic plates. J. Appl. Math. Mech. (Transi, of PMM) 29, 151–166. (21).

    Google Scholar 

  167. Green, A. E., and P. M. Naghdi: 1965 Some remarks on the linear theory of shells. Quart. J. Mech. Appl. Math. 18, 257–276 and 20, 527 (1967). (21).

    MathSciNet  Google Scholar 

  168. Green, A. E., and P. M. Naghdi and R. S. Rivlin: 1965 Directors and multipolar displacements in continuum mechanics. Int. J. Engng. Sci. 2, 611–620. (4, 8).

    MATH  Google Scholar 

  169. Green, A. E., and W. L. Wainwright: 1965 A general theory of a Cosserat surface. Arch. Rational Mech. Anal. 20, 287–308. (4, 5, 6, 8, 9, 13, 16).

    MathSciNet  Google Scholar 

  170. John, F.: 1965 Estimates for the derivatives of the stresses in a thin shell and interior shell equations. Comm. Pure Appl. Math. 18, 235–267. (4, 20).

    MathSciNet  Google Scholar 

  171. Naghdi, P. M.: 1965 A static-geometric analogue in the theory of couple-stress. Proc. Koninkl. Ned. Akad. Wetenschap., Ser. B 68, 29–32. (26).

    MATH  MathSciNet  Google Scholar 

  172. Reissner, E.: 1965 A note on variational principles in elasticity. Int. J. Solids Structures 1, 93–95 and 351. (7).

    Google Scholar 

  173. Tiffen, R., and P. G. Lowe: 1965 An exact theory of plane stress. J. London Math. Soc. 40, 72–86. (12).

    MATH  MathSciNet  Google Scholar 

  174. 1965 Truesdell, C., and W. Noll: The non-linear field theories of mechanics. In Handbuch der Physik, Vol. HI/3 (edit, by S. Flügge). Berlin-Heidelberg-New York: Springer viii+ 602 pp. (2, 4, 8, 11, Chap. D—preliminary remarks, 13, 14, 17).

    Google Scholar 

  175. Cohen, H., and C. N. De Silva: 1966 Nonlinear theory of elastic surfaces. J. Math. Phys. 7, 246–253. (10).

    MATH  Google Scholar 

  176. 1966 Theory of directed surfaces. J. Math. Phys. 7, 960-966. (4, 5, 9, 13).

    Google Scholar 

  177. Green, A. E., and N. Laws: 1966 Further remarks on the boundary conditions for thin elastic shells. Proc. Roy. Soc. London, Ser. A 289, 171–176. (21).

    MathSciNet  Google Scholar 

  178. Koiter, W. T.: 1966 On the nonlinear theory of thin elastic shells. Proc. Koninkl. Ned. Akad. Wetenschap., Ser. B 69, 1–54. (21).

    MathSciNet  Google Scholar 

  179. Kollman, F. G.: 1966 Eine Ableitung der Kompatibilitätsbedingungen in der linearen Schalentheorie mit Hilfe des Riemannschen Krümmungstensors. Ing.-Arch. 35, 10–16. (7).

    Google Scholar 

  180. Laws, N.: 1966 A boundary-layer theory for plates with initial stress. Proc. Cambridge Phil. Soc. 62, 313–327. (21).

    MathSciNet  Google Scholar 

  181. 1966 Naghdi, P. M.: On the differential equations of the linear theory of elastic shells. Proc. Eleventh Int. Congr. Appl. Mech. (Munich, 1964), pp. 262-269. (20, 26).

    Google Scholar 

  182. 1966 Reissner, E.: On the foundations of the linear theory of elastic shells. Proc. Eleventh Int. Congr. Appl. Mech. (Munich, 1964), pp. 20-30. (21 A).

    Google Scholar 

  183. Balaban, M. M., A. E. Green, and P. M. Naghdi: 1967 Simple force multipoles in the theory of deformable surfaces. J. Math. Phys. 8, 1026–1036. (10, 15).

    Google Scholar 

  184. Crochet, M. J.: 1967 Compatibility equations for a Cosserat surface. J. Mécanique 6, 593–600 and 9, 600 (1970). (6).

    MATH  Google Scholar 

  185. Eringen, A. C: 1967 Theory of micropolar plates. Z. Angew. Math. Phys. 18, 12–30. (21).

    Google Scholar 

  186. Green, A. E., and P. M. Naghdi: 1967 The linear theory of an elastic Cosserat plate. Proc. Cambridge Phil. Soc. 63, 537–550 and 922. (6, 7, 9, 13, 16, 24).

    MATH  Google Scholar 

  187. 1967 Micropolar and director theories of plates. Quart. J. Mech. Appl. Math. 20, 183-199. (21).

    Google Scholar 

  188. 1967 Naghdi, P. M.: A theory of deformable surface and elastic shell theory. Proc. Symposium on the theory of shells to honor L. H. Donnell (Houston, 1966), pp. 25-43. (9, 24).

    Google Scholar 

  189. Budiansky, B.: 1968 Notes on nonlinear shell theory. J. Appl. Mech. 35, 393–401. (21).

    Google Scholar 

  190. Cohen, H., and C. N. De Silva: 1968 On a nonlinear theory of elastic shells. J. Mécanique 7, 459–464. (10).

    MATH  Google Scholar 

  191. 1968 Cosserat, E., and F.: Theory of deformable bodies. Translation of [1909, 1]. NASA TT F-11, 561 (Washington, D.C.). Clearinghouse for U.S. Federal Scientific and Technical Information, Springfield, Virginia. (4).

    Google Scholar 

  192. Green, A. E., N. Laws, and P. M. Naghdi: 1968 Rods, plates and shells. Proc. Cambridge Phil. Soc. 64, 895–913-(4, 7, 11, 12, 18).

    MATH  Google Scholar 

  193. Green, A. E., and P.M. Naghdi: 1968 A note on the Cosserat surface. Quart. J. Mech. Appl. Math. 21, 135–139. (15, 25).

    MATH  Google Scholar 

  194. 1968 The linear elastic Cosserat surface and shell theory. Int. J. Solids and Structures 4, 585-592. (6, 9, 13, 16, 24, 25).

    Google Scholar 

  195. 1968 The Cosserat surface. Proc. IUTAM Symposium on mechanics of generalized continua (Freudenstadt and Stuttgart, 1967), pp. 36-48. (9, 15).

    Google Scholar 

  196. Green, A. E., and R. B. Osborn: 1968 Theory of an elastic-plastic Cosserat surface. Int. J. Solids and Structures 4, 907–927. (Chap. D—preliminary remarks).

    MATH  Google Scholar 

  197. 1968 Green, A. E., and W. Zerna: 2nd ed. of [1954, 1]. ix + 457 pp. (4, 7, 11, 12A, 20, 21, 22, 24, Appendix).

    Google Scholar 

  198. Knops, R. J., and L. E. Payne: 1968 Uniqueness in classical elasto-dynamics. Arch. Rational Mech. Anal. 27, 349–355. (26).

    MathSciNet  Google Scholar 

  199. Sensenig, C. B.: 1968 A shell theory compared with the exact three dimensional theory of elasticity. Int. J. Engng. Sci. 6, 435–464. (20).

    MATH  Google Scholar 

  200. Stoker, J. J.: 1968 Nonlinear elasticity. New York: Gordon and Breach, xi + 130 pp. (21).

    MATH  Google Scholar 

  201. Wenner, M. L.: 1968 On torsion of an elastic cylindrical Cosserat surface. Int. J. Solids and Structures 4, 769–776. (24).

    MATH  Google Scholar 

  202. Zerna, W.: 1968 A new formulation of the theory of elastic shells. IASS Bull. 37, 61–76. (7).

    Google Scholar 

  203. Crochet, M. J., and P.M. Naghdi: 1969 Large deformation solutions for an elastic Cosserat surface. Int. J. Engng. Sci. 7, 309–335. (14).

    MATH  MathSciNet  Google Scholar 

  204. Gol’denveizer, A.: 1969 Boundary layer and its interaction with the interior state of stress of an elastic thin shell. J. Appl. Math. Mech. (Transi, of PMM) 33, 971–1001 and 34, 548, (1970). (21).

    Google Scholar 

  205. 1969 Green, A. E., and P. M. Naghdi: Shells in the light of generalized continua. Proc. Second IUTAM Symposium on the theory of thin shells (Copenhagen, 1967), pp. 39-58. (6, 16, 24).

    Google Scholar 

  206. 1969 John, F.: Refined interior shell equations. Proc. Second IUTAM Symposium on the theory of thin shells (Copenhagen, 1967), pp. 1-14. (4, 20).

    Google Scholar 

  207. Tiersten, H. F.: 1969 Linear piezoelectric plate vibrations. New York: Plenum Press, xv+ 212 pp. (20).

    Google Scholar 

  208. Ericksen, J. L.: 1970 Uniformity in shells. Arch. Rational Mech. Anal. 37, 73–84. (14) 2. Green, A. E., and P. M. Naghdi: Non-isothermal theory of rods, plates and shells. Int. J. Solids and Structures 6, 209-244 and 7, 127 (1971). (7, 11, 12, 17, 18, 24) 3., and J. A. Trapp: Thermodynamics of a continuum with internal constraints. Int. J. Engng. Sci. 8, 891-908. (14).

    MATH  MathSciNet  Google Scholar 

  209. Koiter, W. T.: 1970 On the foundations of the linear theory of thin elastic shells, I and II. Proc. Koninkl. Ned. Akad. Wetenschap., Ser. B 73, 169–195. (20).

    MATH  MathSciNet  Google Scholar 

  210. Antman, S. S.: 1971 Existence and nonuniqueness of axisymmetric equilibrium states of nonlinearly elastic shells. Arch. Rational. Mech. Anal. 40, 329–372. (14).

    MATH  MathSciNet  Google Scholar 

  211. Biricikoulu, V., and A. Kalnins: 1971 Large elastic deformations of shells with the inclusion of transverse normalstra in. Int. J. Solids and Structures 7, 431–444. (21).

    Google Scholar 

  212. Crochet, M. J.: 1971 Finite deformations of inextensible Cosserat surface. Int. J. Solids and Structures 7, 383–397. (14).

    MATH  Google Scholar 

  213. Green, A. E., and P. M. Naghdi: 1971 On uniqueness in the linear theory of elastic shells and plates. J. Mécanique 10, 251–261. (26).

    MATH  MathSciNet  Google Scholar 

  214. 1971 On superposed small deformations on a large deformation of an elastic Cosserat surface. J. Elasticity 1, 1-17. (14).

    Google Scholar 

  215. Green, A. E., and M. L. Wenner: 1971 Linear theory of Cosserat surface and elastic plates of variable thickness. Proc. Cambridge Phil. Soc. 69, 227–254. (6, 9, 11, 12, 19, 20, 24).

    MATH  Google Scholar 

  216. Nordgren, R. P.: 1971 A bound on the error in plate theory. Quart. Appl Math. 28, 587–595. (20).

    MATH  MathSciNet  Google Scholar 

  217. Simmonds, J. G.: 1971 An improved estimate for the error in the classical, linear theory of plate bending. Quart. Appl. Math. 29, 439–447. (20).

    MATH  Google Scholar 

  218. Steele, C. R.: 1971 A geometric optics solution for the thin shell equation. Int. J. Engng. Sci. 9, 681–704. (7, 26)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of California, Berkeley, California, USA

    P. M. Naghdi (Professor of Engineering Science)

Authors
  1. P. M. Naghdi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. The Johns Hopkins University, Baltimore, MD, 21218, USA

    C. Truesdell

Rights and permissions

Reprints and permissions

Copyright information

© 1973 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Naghdi, P.M. (1973). The Theory of Shells and Plates. In: Truesdell, C. (eds) Linear Theories of Elasticity and Thermoelasticity. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-39776-3_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-39776-3_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-38853-2

  • Online ISBN: 978-3-662-39776-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation