Effects of Prescribed End Motion on the Dynamic Stability of a Shallow Arch on an Elastic Foundation

Abstract

In this paper we study the effects of elastic foundation on the static and dynamic snap-through of a shallow arch under a point load traveling at a constant speed. The deformation of the arch is expressed in a Fourier series. For static analysis when the moving speed of the point load is almost zero, the first four modes in the expansion are sufficient in predicting the equilibrium positions and the critical loads. Unlike the case without elastic foundation, static snapthrough can occur even when the arch is in another stable (P À 1 ) position before the point load moves onto the arch. In the dynamic case when the moving speed of the point load is significant, the numerical simulation of the response does not converge well, especially long after the point load leaves the arch. However, the total energy of the arch converges quite well when only the first eight modes are used in the Fourier series. This observation allows us to establish a sufficient condition against dynamic snap though, although we are unable to predict precisely, with finite number of modes in the series, at what time it will occur when this sufficient condition is not fulfilled.

Effects of elastic foundation on the snap-through buckling of a shallow arch under a moving point load

“…For a specified h, we can define a b 1 such that there are three one-mode solutions (P 0 ; P þ 1 ; P À 1 ) as long as b < b 1 (Chen and Lin, 2004a)…”

“…On the other hand, for an arch with h > h 2 the critical energy barrier is the strain energy of position P AE 12 (Chen and Lin, 2004a). The two sufficient conditions can then be stated mathematically as…”

“…In the case when the lateral load is applied dynamically instead of in the quasi-static manner, the critical load will be different from the one predicted statically, see Hoff and Bruce (1954), Humphreys (1966), Lock (1966), Hsu (1967Hsu ( , 1968, Hsu et al (1968), Huang and Nachbar (1968), Fulton and Barton (1971), Sundararajan and Kumani (1972), Lo and Masur (1976), Johnson and Mclvor (1978), Johnson (1980), Gregory and Plaut (1982), Donaldson and Plaut (1983), Patricio et al (1998), Xu et al (2002), Lin and Chen (2003), Chen and Lin (2004a), and Chen and Liao (2005). A comprehensive review on the dynamic instability of shallow arches can be found in the book by Simitses (1990).…”

Dynamic Stability of Shallow Arches with the Geometrical Imperfections

“…Although the shallow arch is relatively simple, it is a very useful structural element. Moreover, one can understand the complicated structures that are analogous to shallow arch such as shallow shell and curved plate further, through the research of shallow arch whose structural characteristics have been extensively studied [1][2][3][4][5][6][7][8][9][10][11][12]. When a shallow arch is subject to lateral static load it may become prime buckling or bifurcation buckling under certain conditions.…”