The Effect of Transverse Shear Deformation on the Bending of Elastic Plates

Abstract

A system of equations is developed for the theory of bending of thin elastic plates which takes into account the transverse shear deformability of the plate. This system of equations is of such nature that three boundary conditions can and must be prescribed along the edge of the plate. The general solution of the system of equations is obtained in terms of two plane harmonic functions and one function which is the general solution of the equation Δψ − (10/h2)ψ = 0. The general results of the paper are applied (a) to the problem of torsion of a rectangular plate, (b) to the problems of plain bending and pure twisting of an infinite plate with a circular hole. In these two problems important differences are noted between the results of the present theory and the results obtained by means of the classical plate theory. It is indicated that the present theory may be applied to other problems where the deviations from the results of classical plate theory are of interest. Among these other problems is the determination of the reactions along the edges of a simply supported rectangular plate, where the classical theory leads to concentrated reactions at the corners of the plate. These concentrated reactions will not occur in the solution of the foregoing problem by means of the theory given in the present paper.

A Triangular Plate Bending Element Based on Discrete Kirchhoff Theory with Simple Explicit Expression

“…where φ i , φ xi , φ yi are defined in Equation ( 12), and R i , R xi , R yi and Q i , Q xi , Q yi are defined in Equation (19). We can see that the explicit expression and implementation of the SDKT element in Equation ( 20) are much simpler than the existing nine-DOF triangular plate elements, passing the patch test.…”

“…Most of these elements possess high accuracy and versatility and have been successfully applied to linear or nonlinear analyses of various plate/shell structures [17,18]. In order to overcome the difficulties of the C1 continuity requirement, the Mindlin-Reissner plate theory was proposed for thin and thick plates [19,20], which only required C0 continuity for the displacement functions of the element. However, due to the so-called shear locking phenomenon, these Mindlin-Reissner plate elements usually led to poor results for the analysis of thin plates [21].…”

Some Inconsistencies in the Nonlinear Buckling Plate Theories—FSDT, S-FSDT, HSDT

“…Reissner [ 1 , 2 ] proposed to extend the Timoshenko linear beam theory accounting for a transverse shear effect on the plate theory based on the stress approach 75 years ago. A few years later, Mindlin [ 3 ] developed a displacement-based theory, in which it was assumed that transverse shear stresses were constant through the plate thickness.…”

“…According to this theory, not the angle of rotation in bending but deflection is the primary variable. It imposed simultaneous restrictions on neglecting Reissner boundary effects [ 2 , 15 ]. A number of equations was reduced, and the boundary conditions were altered.…”

Static bending and free vibration analysis of functionally graded porous plates laid on elastic foundation using the meshless method

“…In order to analyze the problem of functionally graded porous plates (FGPP) laid on elastic foundations, one can utilize the two-dimensional shear deformation theories such as the classical plate theory proposed by Kirchhoff [6], the first-order shear deformation theory developed by Mindlin [7], or Reissner [8], and the higher-order shear deformation theories (e.g. [9]), and the refined plate theory [10].…”