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

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

as derived in Horvay-Veluswami [7]. Here g is amplification, τ dimensional time, 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.