Inelastic Behaviour of Plates and Shells pp 225-241 | Cite as

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

## 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(Ωτ_{m};n) = 1 is

as derived in Horvay-Veluswami [7]. Here g is amplification, τ dimensional time, τ_{m} time to failure, Ω natural frequency of membrane oscillation. t = Ω (τ_{m}-τ) is reversed time, i.e., Ωτ_{m} 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 τ_{m}) = 1 (normalization condition), (b) g’ (Ωτ_{m}) = 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 n_{crit} 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

Cylindrical Shell Diagonal Term Impulsive Loading Initial Yield Stress Pade Approximant## Preview

Unable to display preview. Download preview PDF.

## References

- 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.Anderson, D.L.; Lindberg, H.E. “Dynamic pulse buckling of cylindrical shells under transient lateral pressures,” AIAA J. 6 (1968) 569.Google Scholar
- 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.Vaughan, H.; Florence, A.L. “Plastic flow buckling of cylindrical shells due to impulsive loading,” J. Appl. Mech. 92 (1970) 171.CrossRefGoogle Scholar
- 5.Jones, N. “A literature review of the dynamic plastic response of structures,” Parts I and II, Shock & Vibration Digest, 1978.Google Scholar
- 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.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.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.Whittaker, E.T.; Watson, G.N. A Course in Modern Analysis, Cambridge, 1946.Google Scholar
- 10.Stoker, J.J. Nonlinear Vibrations in Mechanical and Electrical Systems, Wiley-Interscience, 1950.MATHGoogle Scholar
- 11.Strutt, M.J.O. Lamesche, Mathieusche u. Verwandte Funktionen in Physik u. Technik, Springer, 1932.Google Scholar
- 12.Hochstadt, H. The Functions of Mathematical Physica, Wiley- -Interscience, 1971.Google Scholar
- 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).Gold, B.; Horvay, G. “Longitudinal heat propagation in three- -phase laminated composites at low exciting frequencies,” J. Appl. Mech. 101 (1979).Google Scholar
- 15.Baker, G.A. Essentials of Padé Approximants, Academic Press, 1975.MATHGoogle Scholar