In this paper we study the dynamic behavior of a shallow arch under a point load Q traveling at a constant speed. Emphasis is placed on finding whether snap-through buckling will occur. In the quasi-static case when the moving speed is almost zero, there exists a critical load Qcr in the sense that no static snap-through will occur as long as Q is smaller than Qcr. In the dynamic case when the point load travels with a nonzero speed, the critical load Qcrd is, in general, smaller than the static one. When Q is greater than Qcrd, there exists a finite speed zone within which the arch runs the risk of dynamic snap-through either while the point load is still on the arch or after the point load leaves the arch. The boundary of this dangerous speed zone can be determined by a more conservative criterion, which employs the concept of total energy and critical energy barrier, to guarantee the safe passage of the point load. This criterion requires the numerical integration of the equations of motion only up to the instant when the point load reaches the other end of the arch.
In this note we show that for a pinned half-sine arch under end couples snap-through buckling will occur unsymmetrically if the initial height of the shallow arch is greater than 6.5466r, where r is the radius of gyration of the cross section. The closed-form expression for the critical couple can be obtained analytically.
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.
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