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Buckling of Simply Supported Bi-Periodic Elastic Columns
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DOI：10.1142/S0219455426501555.png)
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Abstract
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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 N of bi-periodic cells, with N larger than 2. This general expression is corroborated with the buckling values obtained using a direct method for few cells (N = 2 and N = 3 for instance). The behavior of the stability limit for large N 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 N 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 N 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).
Keywords:
Beams
buckling
periodic microstructure
bi-periodic structure
difference equations
eigenvalue problem
asymptotic expansion
gradient beam model
homogenization methods
second-order homogenization
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