Buckling of Simply Supported Bi-Periodic Elastic Columns

In this paper, the buckling of a simply supported stepped periodic column is studied using an analytical method. The column is composed of biperiodic cells of stepped Euler–Bernoulli continuous segments. The deflection solution in each cell can be expressed from the resolution of a fourth-order differential equation. After expressing the continuity conditions between each cell, it is possible to relate the solution of each cell with respect to its neighbors. The differential eigenvalue problem of the bi-periodic structure is converted into a linear difference eigenvalue problem associated to the coefficients of the expressed solution in each cell. A transcendental equation for the buckling load of the continuous biperiodic column is obtained from the resolution of the discrete linear difference eigenvalue problem. This transcendental equation is valid whatever the number [Formula: see text] of bi-periodic cells, with [Formula: see text] larger than 2. This general expression is corroborated with the buckling values obtained using a direct method for few cells ([Formula: see text] and [Formula: see text] for instance). The behavior of the stability limit for large [Formula: see text] values is also specifically studied. It is shown that the bi-periodic column asymptotically converges toward a homogenized Euler–Bernoulli column with equivalent stiffness calibrated from Reuss’s averaging method. More refined beam models are also derived using asymptotic arguments. The buckling load converges toward the one of a gradient beam model for sufficiently large number [Formula: see text] of cells, which can be equivalently derived from a second-order homogenized beam theory. The convergence of this second-order homogenized beam model toward the equivalent homogenized Euler–Bernoulli column (obtained from Reuss’s averaging method) is from below, as also reported for the exact solution of the biperiodic continuous column. A comparison is also carried out for large values of [Formula: see text] with a nonlocal Euler–Bernoulli model, which has the same order of accuracy as obtained from the gradient beam model (second-order homogenized beam model).