A computationally efficient technique to simulate the dynamic response of a beam colliding with rigid obstacles is described in this paper. The proposed method merges three key concepts. First, a low-order discretization scheme that maximizes the number of nodes of the discrete model (where impacts are detected) at the expense of the degree of continuity of the constructed displacement field is used. Second, the constrained problem is transformed into an unconstrained one by formulating the impact by using a Signorini complementarity law involving the impulse generated by the collision and the preimpact and postimpact velocity linked through a coefficient of restitution. Third, Moreau's midpoint time-stepping scheme developed in the context of colliding rigid bodies is used to advance the solution. The algorithm is first validated on the nonimpact problem of a cantilever Rayleigh beam subjected to an impulsive discrete load. Then the problem of a cantilever beam vibrating between two (symmetrically located) stops is analyzed. Both cases of discrete and continuous obstacles are considered, and the numerical predictions are compared with published results or those obtained with a commercial code.

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A post-buckling analysis of a constant or variable length spatial elastica constrained by a cylindrical wall is performed for a first time by adopting an optimal control methodology. Its application in a constrained buckling analysis is shown to be superior when compared to other numerical techniques, as the inclusion of the unilateral constraints is feasible without the need of any special treatment or approximation. Furthermore, the formulation is simple and the optimal configurations of the spatial elastica can be also obtained by considering the minimization condition of the Hamiltonian. We first present the optimal control formulation for the constrained buckling problem of a constant length spatial elastica, including its associated necessary optimality conditions that constitute the Pontryagin's minimum principle. This fundamental constrained buckling problem is used to validate the proposed methodology. The general buckling problem of a variable length spatial elastica is then analyzed that consists of two parts; (1) the solution of the optimal control problem that involves the inserted elastica inside the conduit and (2) the derivation of the buckling load by taking into account the generation of the configurational or Eshelby-like force at the insertion point of the sliding sleeve. A variety of examples are accordingly presented, where the effects of factors, such as the presence of uniform pressure, the clearance of the wall, and the torsional rigidity, on the buckling response of the spatial elastica, are investigated.

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