Boundary conditions at the edge of a thin or thick plate bonded to an elastic support

Abstract

At the clamped edge of a thin plate, the interior transverse deflection ω(x ₁, x₂) of the mid-plane x ₃=0 is required to satisfy the boundary conditions ω=ϖω/ϖn=0. But suppose that the plate is not held fixed at the edge but is supported by being bonded to another elastic body; what now are the boundary conditions which should be applied to the interior solution in the plate? For the case in which the plate and its support are in two-dimensional plane strain, we show that the correct boundary conditions for ω must always have the form

% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakqaabeqaaiaabEhacaqGTaWa% aSaaaeaacaGG0aGaae4vamaaCaaaleqabaGaamOqaaaaaOqaaiaaco% dadaqadaqaaiaacgdacqGHsislcaqG2baacaGLOaGaayzkaaaaaiaa% bIgadaahaaWcbeqaaiaackdaaaGcdaWcaaqaaiaabsgadaahaaWcbe% qaaiaackdaaaGccaqG3baabaGaaeizaiaabIhafaqabeGabaaajaaq% baqcLbkacaGGYaaajaaybaqcLbkacaGGXaaaaaaakiabgUcaRmaala% aabaGaaiinaiaabEfadaahaaWcbeqaaiaadAeaaaaakeaacaGGZaWa% aeWaaeaacaGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGOb% WaaWbaaSqabeaacaGGZaaaaOWaaSaaaeaacaqGKbWaaWbaaSqabeaa% caGGZaaaaOGaae4DaaqaaiaabsgacaqG4bqcaaubaeqabiqaaaqcaa% saaiaacodaaKaaafaajugGaiaacgdaaaaaaOGaeyypa0Jaaiimaiaa% cYcaaeaadaWcaaqaaiaabsgacaqG3baabaGaaeizaiaabIhaliaacg% daaaGccqGHsisldaWcaaqaaiaacsdacqqHyoqudaahaaWcbeqaaiaa% bkeaaaaakeaacaGGZaWaaeWaaeaacaGGXaGaeyOeI0IaaeODaaGaay% jkaiaawMcaaaaacaqGObWaaSaaaeaacaqGKbWaaWbaaSqabeaacaGG% YaaaaOGaae4DaaqaaiaabsgacaqG4bqbaeqabiqaaaqcaauaaKqzGc% GaaiOmaaqcaawaaKqzGcGaaiymaaaaaaGccqGHRaWkdaWcaaqaaiaa% csdacqqHyoqudaahaaWcbeqaaiaabAeaaaaakeaacaGGZaWaaeWaae% aacaGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGObWaaWba% aSqabeaacaGGYaaaaOWaaSaaaeaacaqGKbWaaWbaaSqabeaacaGGZa% aaaOGaae4DaaqaaiaabsgacaqG4bqcaaubaeqabiqaaaqcaasaaiaa% codaaKaaafaajugGaiaacgdaaaaaaOGaeyypa0JaaiimaiaacYcaaa% aa!993A!\[\begin{gathered}{\text{w - }}\frac{{4{\text{W}}^B }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^2 \frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} + \frac{{4{\text{W}}^F }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^3 \frac{{{\text{d}}^3 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}3 \\1 \\\end{array} }} = 0, \hfill \\\frac{{{\text{dw}}}}{{{\text{dx}}1}} - \frac{{4\Theta ^{\text{B}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}\frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} + \frac{{4\Theta ^{\text{F}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^2 \frac{{{\text{d}}^3 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}3 \\1 \\\end{array} }} = 0, \hfill \\\end{gathered}\]with exponentially small error as L/h→∞, where 2h is the plate thickness and L is the length scale of ω in the x ₁-direction. The four coefficients W ^B, W^F, Θ^B, Θ^F are computable constants which depend upon the geometry of the support and the elastic properties of the support and the plate, but are independent of the length of the plate and the loading applied to it. The leading terms in these boundary conditions as L/h→∞ (with all elastic moduli remaining fixed) are the same as those for a thin plate with a clamped edge. However by obtaining asymptotic formulae and general inequalities for Θ^B, W ^F, we prove that these constants take large values when the support is ‘soft’ and so may still have a strong influence even when h/L is small. The coefficient W ^F is also shown to become large as the size of the support becomes large but this effect is unlikely to be significant except for very thick plates. When h/L is small, the first order corrected boundary conditions are

w=0,% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakeaadaWcaaqaaiaabsgacaqG% 3baabaGaaeizaiaabIhaliaacgdaaaGccqGHsisldaWcaaqaaiaacs% dacqqHyoqudaahaaWcbeqaaiaabkeaaaaakeaacaGGZaWaaeWaaeaa% caGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGObWaaSaaae% aacaqGKbWaaWbaaSqabeaacaGGYaaaaOGaae4DaaqaaiaabsgacaqG% 4bqbaeqabiqaaaqcaauaaKqzGcGaaiOmaaqcaawaaKqzGcGaaiymaa% aaaaGccqGH9aqpcaGGWaGaaiilaaaa!5DD4!\[\frac{{{\text{dw}}}}{{{\text{dx}}1}} - \frac{{4\Theta ^{\text{B}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}\frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} = 0,\]which correspond to a hinged edge with a restoring couple proportional to the angular deflection of the plate at the edge.