International Journal of Non-Linear Mechanics 41 (2006) 1065 – 1075 www.elsevier.com/locate/nlm Dynamic buckling of a shallow arch under shock loading considering the effects of the arch shape  N.J. Mallon a , ∗ , R.H.B. Fey a , H. Nijmeijer a , G.Q. Zhang b a Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands b Philips High Tech Campus, Building HTC60, 5656 AG Eindhoven, The Netherlands Received 17 March 2006; received in revised form 24 October 2006; accepted 24 October 2006 Abstract In this paper, the inﬂuence of the initial curvature of thin shallow arches on the dynamic pulse buckling load is examined. Using numerical means and a multi-dof semi-analytical model, both quasi-static and non-linear transient dynamical analyzes are performed. The inﬂuence of various parameters, such as pulse duration, damping and, especially, the arch shape is illustrated. Moreover, the results are numerically validated through a comparison with results obtained using ﬁnite element modeling. The main results are ﬁrstly that the critical shock level can be signiﬁcantly increased by optimizing the arch shape and secondly, that geometric imperfections have only a mild inﬂuence on these results. Furthermore, by comparing the sensitivities of the static and dynamic buckling loads with respect to the arch shape, non-trivial quantitative correspondences are found.  2006 Elsevier Ltd. All rights reserved. Keywords: Dynamic pulse buckling; Semi-analytical models; Shape optimization 1. Introduction Thin-walled structures possess a favorable stiffness-to-mass ratio and are encountered in a wide variety of applications. If such a thin-walled structure is initially curved and is transver- sally loaded above some critical value, the structure may buckle so that its curvature suddenly reverses. This behavior, also known as snap-through buckling [1], is often undesirable. The analysis of structures liable to buckling under static loading is a well established topic in engineering science. However, of- ten thin-walled structures are subjected not only to a static load but also to a distinct dynamic load, such as shock/impact load- ing, step loading or periodic loading. The resistance of struc- tures liable to buckling, to withstand time-dependent loading is often addressed as the dynamic stability of these structures.  This research is supported by the Dutch Technology Foundation STW, applied science division of NWO and the technology programme of the Ministry of Economic Affairs (STW project EWO.5792). ∗ Corresponding author. Tel.: +31 40 247 5730; fax: +31 40 246 1418. E-mail addresses: n.j.mallon@tue.nl (N.J. Mallon), r.h.b.fey@tue.nl (R.H.B. Fey), h.nijmeijer@tue.nl (H. Nijmeijer), g.q.zhang@philips.com (G.Q. Zhang). 0020-7462/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2006.10.017 The corresponding failure mode is often addressed as dynamic buckling and more speciﬁcally as dynamic pulse buckling [2] for the case of pulse loading. Generally, dynamic buckling is related to a large increase in the response resulting from a small increase in some load parameter [3–5]. In the past, many studies have already been performed concerning the dynamic stability of thin-walled structures. Design strategies for such structures under dynamic loading are, however, still lacking. The research described in this paper is intended as a (ﬁrst) step in deriving such design strategies and deals with thin shallow arches under shock loading. The dynamic stability problem of structures can be studied by following an energy based approach [6–8], a numerical approach [9–11] or an experimental approach [12–14]. The energy based approach allows to determine a lower bound for the dynamic buckling load without solving the non-linear equations of motion. However, the established lower bound for the dynamic buckling load by the energy approach can be very conservative [15,16]. Furthermore, the energy based approach does not allow to include the effect of damping rigorously, whereas little damping, as present in all real-life structures, can have a signiﬁcant effect on the dynamic buckling load [9–11].