eesti teaduste
akadeemia kirjastus
SINCE 1952
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of the estonian academy of sciences
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Legacy of Nikolai Alumäe: theory of shells; pp. 139–145
PDF | doi: 10.3176/proc.2015.2.02

Jüri Engelbrecht, Mati Kutser

A short overview on the research of Nikolai Alumäe (1915–1992) in the field of the theory of shells is presented. His brilliant analytical results explaining the stability and transient processes in shells have not lost their importance although obtained in the 1950s and 1960s.


  1. Mushtari, H. M. On stability of the circular cylindrical shell under torsion. Proceedings of the Kazan Aviation Institute, 1933, 2, 3 (in Russian).

  2. Donnell, L. H. Stability of Thin Cylindrical Shells in Torsion. NACA Technical Report, 479. Washington DC, 1933.

  3. Lurye, A. I. General theory of thin elastic shells. Prikl. Mat. Mekh., 1940, 4, 7–34 (in Russian).

  4. Karman, Th. von and Tsien, H. The buckling of thin cylindrical shells under axial compression. J. Aero­naut. Sci., 1941, 8, 303.

  5. Rabotnov, Y. N. Local stability of shells. Dokl. Acad. Sci. USSR, 1946, 52, 111–112 (in Russian).

  6. Reissner, E. On the theory of thin elastic shells. In H. Reissner. Anniversary Volume: Contributions to Applied Mechanics (Edwards, I. W., ed.). Ann Arbor, Michigan, 1949, 231–247.

  7. Goldenveiser, A. L. Equations of the theory of thin shells. Prikl. Mat. Mekh., 1940, 4, 35–42 (in Russian).

  8. Goldenveiser, A. L. Qualitative study of the stress states in thin-walled shells. Prikl. Mat. Mekh., 1945, 9, 463–478 (in Russian).

  9. Goldenveiser, A. L. Momentless theory of shells whose middle surface is a curve of the second order. Prikl. Mat. Mekh., 1947, 11, 285–290 (in Russian).

10. Vlasov, V. Z. The basic differential equations of the general theory of elastic shells. Prikl. Mat. Mekh., 1944, 8, 109–140 (in Russian).

11. Novozhilov, V. V. Theory of Thin Shells. Sudpromgiz, Leningrad, 1951 (in Russian).

12. Ciarlet, P. G. Mathematical Elasticity, Vol. III: Theory of Shells. Studies in Mathematics and Its Applications, North-Holland, Amsterdam, 2000.

13. Ventsel, E. and Krauthammer, T. Thin Plates and Shells. Theory, Analysis, and Applications. Marcel Dekker, Inc., New York, Basel, 2001.

14. Teng, J. G. Buckling of thin shells: recent advances and trends. Appl. Mech. Rev., 1966, 49, 263–274.

15. Alumäe, N. A. Differential equations of the equilibrium of thin elastic shell in postcritical state. Prikl. Mat. Mekh., 1949, 13, 95–106 (in Russian).

16. Alumäe, N. A. On a critical membrane stress state of thin elastic shells. Prikl. Mat. Mekh., 1949, 13, 647–649 (in Russian).

17. Alumäe, N. A. Application of the generalized Castigliano’s variational principle for studies of the post-critical state of thin elastic shells. Prikl. Mat. Mekh., 1950, 14, 647–649 (in Russian).

18. Alumäe, N. A. A variational formula for exploration of thin elastic shells in the post-critical state. Prikl. Mat. Mekh., 1950, 14, 197–202 (in Russian).

19. Vlasov, V. Z. General Theory of Shells and Its Applica­tions in Engineering. Gostekhizdat, Moscow–Lenin­grad, 1949 (in Russian).

20. Alumäe, N. A. An investigation of the post-critical stage of thin elastic shells by the method of power series of the small parameter. In Nõukogude teaduse arengust Eesti NSV-s. 1940–1950. Eesti Riiklik Kirjastus, Tallinn, 1950, 341–353 (in Russian).

21. Panov, D. Y. On large deflections of the circular slightly corrugated membranes. Prikl. Mat. Mekh., 1941, 5, 303–318 (in Russian).

22. Panov, L. Y. On stability of the heated bimetallic shell. Prikl. Mat. Mekh., 1947, 11, 603–310 (in Russian).

23. Feodosiev, V. I. Large displacements and stability of a circular membrane with fine corrugations. Prikl. Mat. Mekh., 1945, 9, 389–412 (in Russian).

24. Feodosiev, V. I. Elastic Elements in the Precise Instrument Making. Oborongiz, Moscow, 1949 (in Russian).

25. Alumäe, N. A. On the theory of axisymmetric deforma­tions of shells of rotation with finite deformations. Prikl. Mat. Mekh., 1952, 16, 419–428 (in Russian).

26. Federhofer, K. Zur Stabilität der Katenoidschale. S.-B. Akad. Wiss. Wien, Math.‑Naturw., 1939, K1, Abt. IIa, 148, 67–80.

27. Alumäe, N. A. On the critical value of the axisymmetric membrane stress-state of a catenoid shell. Prikl. Mat. Mekh., 1952, 16, 419–428 (in Russian).

28. Alumäe, N. A. On determination of the state of equilibrium of the circular shell under axisymmetric loading. Prikl. Mat. Mekh., 1953, 17, 517–528 (in Russian).

29. Alumäe, N. A. Critical loading of the long cylindrical circular shell under torsion. Prikl. Mat. Mekh., 1954, 18, 24–34 (in Russian).

30. Goldenveiser, A. L. Theory of the Thin Elastic Shells. Gostekhizdat, 1953 (in Russian).

31. Lurye, A. I. On equations of the general theory of elastic shells. Prikl. Mat. Mekh., 1950, 14, 558–560 (in Russian).

32. Alumäe, N. A. On the presentation of fundamental correla­tions of the nonlinear theory of shells. Prikl. Mat. Mekh., 1956, 20, 136–139 (in Russian).

33. Alumäe, N. A. Determination of the critical pressure on the shell outlined on the surface of the hyperboloid of one sheet. Prikl. Mat. Mekh., 1956, 20, 223–235 (in Russian).

34. Alumäe, N. A. On the asymptotic integration of equations of the statical stability of the conical shell of revolu­tion. Prikl. Mat. Mekh., 1957, 21, 83–88 (in Russian).

35. Alumäe, N. A. Stability of the equilibrium of the helicoidal shell. Prikl. Mat. Mekh., 1957, 21, 823–826 (in Russian).

36. Alumäe, N. On the fundamental system of integrals of the equation of small steady axisymmetrical vibrations of the elastic conical shell. Eesti NSV Tead. Akad. Toim. Füüs.-Matem. Tehn., 1960, 9, 3–15 (in Russian).

37. Grigolyuk, E. I. On vibrations of the flat circular cylindrical panel having finite deflections. Prikl. Mat. Mekh., 1955, 19, 136–139 (in Russian).

38. Grigolyuk, E. I. Nonlinear vibrations and stability of flat rods and shells. Izv. Akad. Nauk SSSR, OTN, Mekh. Mashinostr., 1955, 3, 36–68 (in Russian).

39. Agamirov, V. L. and Volmir, A. S. Behaviour of cylindrical shells dynamical all-around pressure or axial loading. Izv. Akad Nauk SSSR, OTN, Mekh. Mashinostr., 1959, 3, 78–83 (in Russian).

40. Alumäe, N. On the decomposition of the stress state in axisymmetrical problems of motion for circular cylindrical shell. Eesti NSV Tead. Akad. Toim. Füüs.-Matem. Tehn., 1961, 10, 171–181 (in Russian).

41. Herrmann, G. and Mirsky, I. Three-dimensional and shell-theory analysis of axially symmetric motions of cylinders. J. Appl. Mech., 1956, 23, 563–568.

42. Alumäe, N. and Poverus, L. Transient stresses in a semi-infinite elastic circular cylindrical shell under non-axial loading. Eesti NSV Tead. Akad. Toim. Füüs.-Matem. Tehn., 1961, 12, 13–23 (in Russian).

43. Alumäe, N. Discontinuities in accelerations of a spherical shell caused by the plane pressure wave. In Proc. 6th All-Union Conference on Theory of Shells and Plates, Baku. 1966, 44–47 (in Russian).

44. Teng, J. G. and Rotter, J. M. (eds). Buckling of Thin Metal Shells. Spon Press, London, 2004.

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