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10.3: Effect of Boundary Conditions

  • Page ID
    21534
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    The unloaded edges of rectangular plates can be either simply supported (ss), clamped (c) or free. (The sliding boundary conditions will convert the eigenvalue problem into the equilibrium problem and therefore are not considered in the buckling analysis of plates). The loaded edges could be either simply supported or clamped. This gives rise to ten different combination. The buckling coefficient is plotted against the plate aspect ratio \(a/b\) for all these combinations in Figure (10.4.1). It is seen that the lowest buckling coefficient with \(m = 1\) corresponds to a simply supported plate on three edges and free on the fourth edge.

    An approximate analytical solution for the case “E” was derived by Timoshenko and Gere in the form

    \[k_c = 0.456 + \left(\frac{b}{a}\right)^2\]

    For example \(k_c = 0.706\) for \(a/b = 2\), which is very close to the value that could be read off from Figure (10.4.1). An angle element, shown in Figure (10.4.2) is composed of two plates that are simply supported along the common edge and free on the either edges. Both plates rotate by the same amount at the common edges so that no edge restraining moment is developed. This corresponds to a simply supported boundary conditions.

    In a similar way it can be proved that the prismatic square column consists of four simply supported long rectangular plates. Upon compression, the buckling pattern has a form shown in Figure (10.4.3). Again, there are no relative rotations at the intersection line of any of the neighboring plates ensuring the simply supported boundary condition along four edges.

    Another very practical case is shear loading. For example “I” beams with a relatively high web or girders may fail by shear buckling, Figure (10.4.4), in the compressive side when subjected to bending.

    The solution to the shear buckling is much more complicated than in the previous cases of compressive buckling. The general form of the solution is still given by Equation (??) but there is no simple closed form solution for the buckling coefficient. An approximate solution for \(k_c\), derived by Timoshenko and Gere has the form

    \[k_c = 5.35 + 4\left(\frac{b}{a}\right)^2\]

    For a square plate the buckling coefficient is 9.35 while for an infinitely long plate, \(a \ll b\) it reduces to 5.35. Loading the plate in the double shear experiment for beyond the elastic buckling load produces a set of regular skewed dimples seen in Figure (10.4.5).


    This page titled 10.3: Effect of Boundary Conditions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Tomasz Wierzbicki (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.