Abstract
The stiffness model of the finite element is applied to the Kirchhoff-love closed-form plate buckling; buckling is always in focus in plate assemblages. The useful Eigen-value solutions are unable to separate a square plate from a much weaker long one in the most commonly-used all-simply supported plate (SSSS), among others. Spring-values of the Kirchhoff-Love plate are sought; once found, displacement-factors can be determined. Comparative displacements allow an easier and better evaluation of buckling-factors, pure-shear, vibration and so are termed “buckling-displacement-factors”. In testing, many plates in mixed boundary conditions are evaluated for displacement assisted buckling-solutions, first. The displacement-factors made from fundamental Eigen-vectors, in a single-pass, are found to be within about one-percent of known elastic values. It is found that the Kirchhoff-Love plate spring and the finite-element spring, demonstrated, here, in the assemblage of beam-elements, are equivalent from the results. In either case, stiffness is first assembled, ready for any loading—transverse, buckling, shear, vibration. The simply-supported plate draws the only exact vibration solution, and so, in an additional new effort, all other results are calibrated from it; direct vibration solutions are made for comparison but such results are, hardly, better. In the process, interactive Kirchhoff-Love plate-field-sheets are presented, for design. It is now additionally demanded that the solution Eigen-vector be developable into a recognizable deflection-factor. A weaker plate cannot possess greater buckling strength, this is a check; to find stiffness the deflection-factor must be exact or nearly so. Several examples justify the characteristic buckling displacement-factor as a new tool.
Highlights
Relative displacement-factors are employed in very accurate solutions for buckling, pure-shear and vibration
2) For self-checking, it is demanded the solution buckling-mode be developable into recognizable elastic deflection-factor; for example, for the Euler bar Δ = 0.0103077 checking with the Euler-Bernoulli beam of Δ = 0.01302; for the “CCCC” plate, Δ = 0.00128 checking with Timoshenko’s, Δ = 0.00126; etc
3) Tracking buckling by displacement-factors has brought to light existing large errors in some existing solutions; this study confirms that the strength of the “SSSS” plate in uni-axial compression decreases from 4-units at a/b = 1 to 2.6 units at a/b→∞, and not a constant 4-units; the difference is huge
Summary
In the area of plate bending and displacement, Timoshenko and Krieger [1], remains a veritable source of reference including the contribution of many other expert-authors. The complexities in buckling solutions quickly become apparent in the very first problem in “SSSS” where the square-plate and another “1. The expected buckling-strength versus aspect-ratio of the “SSSS” plate in uni-axial compression is expected to be “Graph-B” rather than the stronger existing graph-A. The buckling load of the “SSSS-plate” cannot have the same result between the aspect-ratios of 1.0 and “∞” as exhibited in existing monographs, as handed down since [1] [2]. In Johnarry and Ebitei, [7] [8], buckling was tracked by displacement and the Euler-type Curve-B, was achieved; the buckling strength of infinitely long “SSSS-plate” reduced significantly below the “4.0” literature value, [1] [2]. Similar improvements were found in other cases and especially the pure-shear cases [8] [9] [10]
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