Dynamic Buckling of a Rigid-Plastic Cylindrical Shell: A Second Order Differential Equation Subject to Four Boundary Conditions

  • Karen Zak Benbury
  • M. A. Veluswami
  • G. Horvay
Conference paper
Part of the International Union of Theoretical and Applied Mechanics book series (IUTAM)

Summary

In the case of a Loss-Of-Coolant Accident an impulsive pressure acts on the core-support-barrel within the nuclear reactor, and the danger arises that the out-of-round shell will buckle. The dimensionless equation governing the growth g(t) of initial imperfection g(Ωτm;n) = 1 is

$$ {\text{ E}}{\mkern 1mu} \{ {\text{g}}{\mkern 1mu} ({\text{t}};{\text{n}}){\mkern 1mu} \} \equiv {\mkern 1mu} {\text{g''}}{\mkern 1mu} {\mkern 1mu} - {\text{pg'}}{\mkern 1mu} \cos {\text{ec}}{\mkern 1mu} {\mkern 1mu} t - qg - r = 0$$

as derived in Horvay-Veluswami [7]. Here g is amplification, τ dimensional time, τm time to failure, Ω natural frequency of membrane oscillation. t = Ω (τm-τ) is reversed time, i.e., Ωτm represents the instant of initiating the impulse, t = 0 the time of collapse. p(n), q(n), r(n) are functions of material properties, geometry (i.e., dimensions of the shell), and the buckling mode n; r in addition depends also on the loading condition. The mathematical problem: to so solve E = 0 that the boundary conditions be satisfied: (a) g(Ω τm) = 1 (normalization condition), (b) g’ (Ωτm) = 0 (zero initial velocity), (c) g’ (0) = 0 (zero final velocity), (d) g(0) = peak value (solve the E=0 problem vs n and then select the value ncrit for which g reaches peak value). Note that conditions (c) and (d) are redundant, because when peak value is reached, the velocity is zero. Three methods of solution are considered: I. Power series expansion in t (in conjunction with Padé approximants), exceedingly rapid convergence is observed; II. Cosine series solution, and its pair obtained by the variation of parameters method; III. Pseudo-Hell analysis. But only I is carried to completion, including a numerical confirmation that for the aluminum shell of Vaughan-Florence [4], the results coincide with those of Horvay-Veluswami-Stockton [6], based on time-consuming (and eo ipso, expensive) numerical forward integration, which also requires a passage to ∞ in the number N of integration steps. Because of the excellence of I, the traditional method II is stopped short of numerical calculations. While III constitutes a fascinating counterpart of the Hill-Mathieu theory, it would primarily reveal behavior of the solution of E = 0 at large negative values of t, whereas the physical problem terminates at t = 0; for this reason no effort was made to carry out the analysis III, only its principal features are outlined.

Keywords

Assure Congo 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abrahamson, G.R.; Goodier, J.N. “Dynamic plastic flow buckling of a cylindrical shell from uniform radial impulse,” Proc. 4th U.S. Nat’l. Cong. Applied Mech. 2 (1962) 939.Google Scholar
  2. 2.
    Anderson, D.L.; Lindberg, H.E. “Dynamic pulse buckling of cylindrical shells under transient lateral pressures,” AIAA J. 6 (1968) 569.Google Scholar
  3. 3.
    Florence, A.L.; Vaughan, H. “Dynamic plastic buckling of short cylindrical shells due to impulsive loading,” Internat’l J. Solids Structures 4 (1968) 741.CrossRefMATHGoogle Scholar
  4. 4.
    Vaughan, H.; Florence, A.L. “Plastic flow buckling of cylindrical shells due to impulsive loading,” J. Appl. Mech. 92 (1970) 171.CrossRefGoogle Scholar
  5. 5.
    Jones, N. “A literature review of the dynamic plastic response of structures,” Parts I and II, Shock & Vibration Digest, 1978.Google Scholar
  6. 6.
    Horvay, G.; Veluswami, M.A.; Stockton, F.D. “Dynamic plastic buckling of shells: a reconsideration of the Vaughan- -Florence analysis,” Proc. 5th SMIRT Conf., Berlin (1979) paper L4/8.Google Scholar
  7. 7.
    Horvay, G.; Veluswami, M.A. “Extension of the Vaughan- -Florence analysis of dynamic buckling of a rigid-plastic cylindrical shell,” Proc. SECTAM Conference, Knoxville (1980).Google Scholar
  8. 8.
    Veluswami, M.A.; Horvay, G. “Plastic buckling of cylindrical shells under axially varying dynamic pressure loading,” Proc. 6th SMIRT Conference, Paris (1981) paper L14/2.Google Scholar
  9. 9.
    Whittaker, E.T.; Watson, G.N. A Course in Modern Analysis, Cambridge, 1946.Google Scholar
  10. 10.
    Stoker, J.J. Nonlinear Vibrations in Mechanical and Electrical Systems, Wiley-Interscience, 1950.MATHGoogle Scholar
  11. 11.
    Strutt, M.J.O. Lamesche, Mathieusche u. Verwandte Funktionen in Physik u. Technik, Springer, 1932.Google Scholar
  12. 12.
    Hochstadt, H. The Functions of Mathematical Physica, Wiley- -Interscience, 1971.Google Scholar
  13. 13.
    Horvay, G.; Gold, B.; Kaczenski, E.S. “Longitudinal heat propagation in three-phase laminated composites at high exciting frequencies,” J. Heat Transfer 100 (1978) 281.CrossRefGoogle Scholar
  14. 14).
    Gold, B.; Horvay, G. “Longitudinal heat propagation in three- -phase laminated composites at low exciting frequencies,” J. Appl. Mech. 101 (1979).Google Scholar
  15. 15.
    Baker, G.A. Essentials of Padé Approximants, Academic Press, 1975.MATHGoogle Scholar

Copyright information

© Springer, Berlin Heidelberg 1986

Authors and Affiliations

  • Karen Zak Benbury
    • 1
  • M. A. Veluswami
    • 2
  • G. Horvay
    • 3
  1. 1.Dept MathU.S. Naval AcademyAnnapolisUSA
  2. 2.Dept MEIndian Inst. of Tech.MadrasIndia
  3. 3.Dept MAENorth Carolina State UniversityRaleighUSA

Personalised recommendations