Theories of elastic plates

1 Basic relations in theories of elastic plates.- I Isotropic rectangular plates.- 2 Differential equations of the component theory.- 2.1 Summary of basic equations.- 2.2 Transformation of basic equations for resultant quantities.- 2.3 Some special cases.- 3 The Reissner plate theory.- 3.1 Summary of governing equations.- 3.2 Neglecting transverse contraction in the Reissner theory.- 4 Refined theory of Hencky.- 4.1 Summary of governing equations.- 4.2 Modification of the Hencky theory based on more accurate displacement functions.- 5 Refined theory of Kromm.- 5.1 Bending state.- 5.1.1 Cases p = const., p = 0. Comparison with the component theory.- 5.1.2 Case of a general harmonic load.- 5.1.3 Particular solution for a general non-harmonic load.- 5.1.4 Comparison with the second approximation of the component theory.- 5.2 Antisymmetric stress state of the plate without deflections.- 5.2.1 Comparison with the third approximation of the component theory.- 5.3 Satisfaction of Kromm’s equations in their integral form.- 6 Generalized theory.- 6.1 Summary of governing equations.- 6.2 Process of solution.- 6.3 Comparison with other refined theories.- 7 Boundary conditions.- 7.1 Boundary conditions in refined theories.- 7.1.1 Free edge.- 7.1.2 Simply supported edge.- 7.1.3 Elastically supported edge.- 7.1.4 Built-in edge.- 7.1.5 Elastically built-in edge.- 7.1.6 Continuous plate.- 7.1.7 Elastically supported continuous plate.- 7.1.8 Continuous plate on elastically rotating supports.- 7.1.9 Curvilinear edge.- 7.2 Boundary conditions leading to integration problems of the fourth order.- 7.2.1 Basic relations for resultant quantities of the second approximation of the component theory.- 7.2.2 Basic relations for particular solutions defined by the generalized theory.- 7.2.3 Free edge.- 7.2.4 Simply supported edge.- 7.2.5 Built-in edge.- 8 Bending of long rectangular plates to a cylindrical surface.- 8.1 General solution following from the component theory.- 8.1.1 Cantilever.- 8.1.2 Strip with simply supported edges.- 8.1.3 Strip with built-in edges.- 8.1.4 Strip with one edge simply supported and other edge built-in.- 8.1.5 Continuous plate strip.- 8.2 General solution following from the generalized theory.- 8.2.1 Cantilever.- 8.2.2 Strip with simply supported edges.- 8.2.3 Strip with built-in edges.- 8.2.4 Strip with one edge simply supported and other edge built-in.- 8.2.5 Continuous plate strip.- 9 Solution of the boundary value problem for a rectangular plate.- 10 Plate with two opposite edges simply supported.- 10.1 Bending by moments distributed along one simply supported edge.- 10.1.1 Deformation symmetrical with respect to both coordinate axes.- 10.1.2 Deformation symmetrical with respect to the x-axis and antisym- metrical to the y-axis.- 10.1.3 Deformation symmetrical with respect to the y-axis and antisymmetri- cal to the x-axis.- 10.1.4 Deformation antisymmetrical with respect to both coordinate axes.- 10.2 Bending by moments distributed along one free edge.- 10.2.1 Deformation symmetrical with respect to both coordinate axes.- 10.2.2 Deformation symmetrical with respect to the y-axis and antisymmetrical to the x-axis.- 10.2.3 Deformation symmetrical with respect to the x-axis and antisymmetrical to the y-axis.- 10.2.4 Deformation antisymmetrical with respect to both coordinate axes.- 10.3 Bending by shearing forces distributed along one free edge.- 10.3.1 Deformation symmetrical with respect to both coordinate axes.- 10.3.2 Deformation symmetrical with respect to the y-axis and antisymmetrical to the x-axis.- 10.3.3 Deformation symmetrical with respect to the x-axis and antisymmetrical to the y-axis.- 10.3.4 Deformation antisymmetrical with respect to both coordinate axes.- 11 Free rectangular plate.- 11.1 Bending by moments distributed along the edges.- 11.1.1 Deformation symmetrical with respect to both coordinate axes.- 11.1.1.1 Influence of the boundary moment distribution and of the thickness on deflections of a square plate. Numerical comparison.- 11.1.2 Deformation symmetrical with respect to the x-axis and antisymmetrical to the y-axis.- 11.1.3 Deformation symmetrical with respect to the y-axis and antisymmetrical to the x-axis.- 11.2 Bending by twisting moments distributed along the edges.- 11.2.1 Deformation antisymmetrical with respect to both coordinate axes.- 11.2.1.1 Influence of the distribution of boundary twisting moments and of the plate thickness.- 11.2.2 Deformation symmetrical with respect to both coordinate axes.- 11.2.2.1 Influence of plate thickness on deflections.- 12 Simply supported plate subjected to boundary bending moments.- 12.1 Deformation symmetrical with respect to both coordinate axes.- 12.1.1 Influence of plate thickness on the maximum deflection. Comparison with the classical theory.- 12.2 Deformation symmetrical with respect to the x-axis and antisymmetrical to the y-axis.- 12.3 Deformation symmetrical with respect to the y-axis and antisymmetrical to the x-axis.- 12.4 Deformation antisymmetrical with respect to both coordinate axes.- 13 Completely simply supported plate under distributed loading.- 13.1 Solution according to the component theory.- 13.1.1 Square plate. Influence of the plate thickness and of the number of terms considered.- 13.2 Solution according to the generalized theory.- 13.2.1 Square plate.- 14 Completely built-in plate under distributed loading.- 14.1 Solution according to the component theory.- 14.1.1 Square plate. Influence of the plate thickness. Comparison with the classical theory.- 14.2 Solution according to the generalized theory.- 15 Plate with two opposite edges simply supported and the other two edges free under a continuous load.- 15.1 Solution according to the component theory.- 15.2 Solution according to the generalized theory.- 16 Plate with two opposite edges simply supported and the other two edges built-in under a continuous load.- 16.1 Solution according to the component theory.- 16.2 Solution according to the generalized theory.- II Orthotropic rectangular plates.- 17 Differential equations of the component theory. Case H= (KxKy)½.- 17.1 Summary of governing equations.- 17.2 Transformation of governing equations for resultant quantities.- 17.3 Transformation of coordinates.- 18 Generalized theory.- 18.1 Summary of governing equations.- 18.2 Governing equations for the case H = (KxKy)½.- 19 Refined theory developed by K. Girkmann and R. Beer.- 19.1 Summary of governing equations.- 19.2 Neglect of transverse contraction. Comparison with the generalized theory.- 19.3 Governing equations for the ease H = (KxKy)½ Comparison with the component theory.- 20 General solution of the boundary value problem according to the component theory.- 21 Orthotropic rectangular plate under a continuous load.- 21.1 Cantilever.- 21.2 Plate with one edge built-in, the opposite edge free, and the other edges simply supported.- 21.3 Plate with three edges built-in and one edge free.- 21.4 Plate with one edge built-in, the opposite edge free, and the other edges elastically built-in.- 21.5 Plate with three edges simply supported and one edge free.- 21.6 Plate with one edge simply supported, the opposite edge free, and the other edges built-in.- III Isotropic circular plates.- 22 Differential equations of the component theory in cylindrical coordinates.- 22.1 Summary of governing equations.- 22.2 Transformation of governing equations for resultant quantities.- 22.3 Alternate method of derivation of governing equations.- 23 Generalized theory of circular plates.- 23.1 Procedure of solution.- 23.2 Case ?rp = 0.- 24 Rotationally symmetrical bending.- 24.1 General solution of governing equations.- 24.2 Particular solutions for some continuously distributed loads.- 25 Circular plates under rotationally symmetrical loads.- 25.1 Bending by moments uniformly distributed along the edge.- 25.2 Uniformly loaded circular plate.- 25.2.1 Simply supported edge.- 25.2.2 Absolutely built-in edge.- 25.2.3 Elastically built-in edge.- 25.3 Circular plate under a partial uniform load.- 25.3.1 Simply supported edge.- 25.3.2 Built-in edge.- 26 Annular plates under rotationally symmetrical loads.- 26.1 Simply supported outer edge.- 26.2 Both edges simply supported.- 26.3 Both edges built-in.- 27 General solution of governing equations of the component theory for loads distributed according to the law p?(?) sin n ?.- 27.1 Formulae for differentiation of modified Bessel functions.- 27.2 Case n = 1.- 27.3 Case n = 2.- 28 Antisymmetrical bending of a circular plate under the load ?? sin ?.- 28.1 Simply supported edge.- 28.2 Built-in edge.- 28.3 Elastically built-in edge.- IV Circular plates on elastic foundation under rotationally symmetrical loads.- 29 Governing equations of the component theory.- 29.1 Introduction of a new dimensionless variable.- 29.2 General solution of the governing equation.- 29.2.1 Case x 1.- 29.3 Two-parameter elastic foundation.- 30 Auxiliary formulae for Bessel functions.- 30.1 Functions of complex arguments.- 30.2 Functions of pure imaginary arguments.- 31 Functions expressing the general solution of the homogeneous problem.- 31.1 Case x < 1.- 31.2 Case x = 1.- 31.3 Case x > 1.- 32 Fundamental solution.- 32.1 Case x < 1.- 32.1.1 Infinite plate under a concentrated load.- 32.1.2 Load uniformly distributed along a circle.- 32.1.3 Load uniformly distributed over an annular surface.- 32.1.4 Load uniformly distributed over a circular surface.- 32.2 Case x = 1.- 32.2.1 Infinite plate under a concentrated load.- 32.2.2 Load uniformly distributed along a circle.- 32.2.3 Load uniformly distributed over an annular surface.- 32.2.4 Load uniformly distributed over a circular surface.- 32.3 Case x > 1.- 32.3.1 Infinite plate under a concentrated load.- 32.3.2 Load uniformly distributed along a circle.- 32.3.3 Load uniformly distributed over an annular surface.- 32.3.4 Load uniformly distributed over a circular surface.- 32.4 Effect of the parameter x on deflection of an infinite plate under a concentrated load.- 33 Compensating solution.- 33.1 Circular plate.- 33.1.1 Free edge.- 33.1.1.1 Case x < 1.- 33.1.1.2 Case x = 1.- 33.1.1.3 Case x > l.- 33.1.2 Simply supported edge.- 33.1.2.1 Case x < 1.- 33.1.2.2 Case x = 1.- 33.1.2.3 Case x > 1.- 33.1.3 Built-in edge.- 33.1.3.1 Case x < 1.- 33.1.3.2 Case x = 1.- 33.1.3.3 Case x > 1.- 33.2 Infinite plate with a circular hole.- 33.3 Annular plate.- 34 Numerical example.- 34.1 Case x = 0.- 34.2 Case x = 0.17365.- 34.3 Case x= 1.0.- 34.4 Case x = 2.125.- Author index.