Buckling of multiply connected bar-chain and its associated continualized nonlocal model

This paper investigates the buckling problem of a multiply connected bar-chain (MCB) system. Unlike the Hencky bar-chain that comprises rigid bars connected to each other by rotational springs, this MCB model has rigid bars connected by both direct and indirect neighbouring rotational springs that simulate the long range interactions between bars. The buckling loads of the MCB with two-neighbour and N-neighbour interactions are analytically obtained for simply supported, clamped-clamped and clamped-free boundary conditions. By continualizing the discrete equations associated with MCB, a nonlocal continuum model is constructed to predict the buckling behavior of nonlocal bars with short and long range interactions. It is shown that this continualized nonlocal model (CNM) is equivalent to a stress gradient type of Eringen's nonlocal media. By comparing the solutions between CNM and MCB, it is shown that CNM is a simple nonlocal continuum model that is able to capture the scale effect of a generalized lattice that includes multiple interactions among its components. The closed form expression of the length scale of CNM is derived in terms of the rotational spring stiffnesses of MCB.