Abstract
Sixteen-decade-old problem of Poisson-Kirchhoff’s boundary conditions paradox is resolved in the case of isotropic plates through a theory designated as “Poisson’s theory of plates in bending.” It is based on “assuming” zero transverse shear stresses instead of strains. Reactive (statically equivalent) transverse shear stresses are gradients of a function (in place of in-plane displacements as gradients of vertical deflection) so that reactive transverse stresses are independent of material constants in the preliminary solution. Equations governing in-plane displacements are independent of the vertical (transverse) deflectionw0(x,y). Coupling of these equations withw0is the root cause for the boundary conditions paradox. Edge support condition onw0does not play any role in obtaining in-plane displacements. Normally, solutions to the displacements are obtained from governing equations based on the stationary property of relevant total potential and reactive transverse shear stresses are expressed in terms of these displacements. In the present study, a reverse process in obtaining preliminary solution is adapted in which reactive transverse stresses are determined first and displacements are obtained in terms of these stresses. Equations governing second-order corrections to preliminary solutions of bending of anisotropic plates are derived through application of an iterative method used earlier for the analysis of bending of isotropic plates.
Highlights
Kirchhoff ’s theory [1] and first-order shear deformation theory based on Hencky’s work [2] abbreviated as FSDT of plates in bending are simple theories and continuously used to obtain design information
Kirchhoff ’s theory consists of a single variable model in which in-plane displacements are expressed in terms of gradients of vertical deflection w0(x, y) so that zero face shear conditions are satisfied. w0 is governed by a fourth-order equation associated with two edge conditions instead of three edge conditions required in a 3D problem
It is shown that the second-order correction to w0(x, y) by either Reissner’s theory or FSDT corresponds to approximate solution of a torsion problem [5]
Summary
Kirchhoff ’s theory [1] and first-order shear deformation theory based on Hencky’s work [2] abbreviated as FSDT of plates in bending are simple theories and continuously used to obtain design information. Kirchhoff ’s theory consists of a single variable model in which in-plane displacements are expressed in terms of gradients of vertical deflection w0(x, y) so that zero face shear conditions are satisfied. The condition ωz = 0 decoupling the bending and torsion problems is satisfied in Kirchhoff ’s theory If this condition is imposed in FSDT, sum of the strains (εx + εy) in the isotropic plate is governed by a second-order equation due to applied transverse loads. Reactive transverse shear stresses and thickness-wise linear strain εz form the basis for resolving the paradox and for obtaining higher order corrections to the displacements. Equations governing second-order corrections to the preliminary solutions are derived through the application of an iterative method used earlier [7] for the analysis of bending of isotropic plates
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