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

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.

A general condition for the existence of unconnected equilibria for symmetric arches

“…Masur [3] directly solved the differential equations analytically and made a special effort to differentiate symmetric and asymmetric snap-through. These pioneering studies have been extended by many researchers to examine the influence of elastic boundary constraints on critical equilibrium states [4][5][6][7][8][9], to characterize the sensitivity of buckling loads to geometric and load imperfections [10][11][12][13][14][15][16][17], to derive exact solutions for critical loads without truncating the Fourier series [18,19], to incorporate thermal effects in nonlinear buckling analyses [20][21][22][23], and to study the effects of pre-buckling deformations, initial shapes and loading on the nonlinear stability responses [23][24][25][26][27][28][29]. In these investigations, the snap-through buckling loads were obtained but few efforts were made to characterize the post-buckling responses.…”

Stability of a half-sine shallow arch under sinusoidal and step loads in thermal environment

“…In most cases, shallow arches become elastically unstable when the lateral load reaches a critical value (Chen and Li, 2006). This means that a large deformation could be observed while the material remains elastic.…”

Stability of shallow arches under constant load

“…This type of research have also been done in engineering. In fact, stability of the shallow arch from an engineering point of view is studied in [4,5,8,15,16,17,21], and in their references. This work dealt with the subject of chaotic motion, global behavior, accurate solutions, dynamic critical load, internal resonance, and stability with various boundary conditions and load type.…”