Side Of The Supporting Rectangular Plate Bending Accurate Solution

The elastic thin plate is widely used in each domain of engineering, It has high theoretical significance and certain practice to research a theoretical solution to the plate bending, especially an unified solution or an accurate solution which adapts to various boundary conditions, load conditions and displacement of the boundary. In this paper, we studied the accurate bending solution to rectangular plates with one supported edge, which is a sub-item of the accurate solution of rectangular plate bending.Various classic solutions and their limitations were summarized in this thesis, and especially expounded the basic thought of the unified solution. According to the completeness on calculating conditions of the corner point, it divided the plate bending into the generalized statically determinate bending and indeterminate bending. If the plate has supported corner point and the reverse force can be determined by the static equilibrium equation, it belongs to the generalized statically determinate bending; the others belong to the generalized indeterminate bending. As the calculating condition is complete, the plate bending can be solved directly by using the equilibrium differential equation and the boundary conditions. For the latter the superposition method can be used. When solving the generalized statically determinate bending, the deflection expression consists of a homogeneous part W₁ and a particular part W ₂, which represent the homogeneous and particular solution respectively. The common form of W₁ is a bi-directional single trigonometric series that contains eight undetermined constants to reflect the bi-directional bending deformation. The trigonometric series should be complete and be suitable to the deformation resulting from the boundary displacement. The homogeneous solution W₁ should also satisfy the displacement conditions of the column support, the concentrated force condition at the column support, and reflect the boundary's linear displacement character that resulted form the displacement at endpoint of supported edge. The generalized statically determinate bending can be divided into seven kinds of rectangular plates according to the boundary's approximation. The particular solution W2 reflects the loads on the plate and concentrated fore at the free corner point. The form of particular solution may use duple trigonometric series or multinomial when loads operate on the plate, it can only use multinomial when concentrated force operates on the corner point.Though the unified solution has made a big improvement than those classic solutions, it still has some drawbacks. The accurate solution founded one set of completely new solution system based on the unified one. Firstly, it adopted more scientific reason thought to divide the generalized statically determinate bending and indeterminate bending. It considers that the equilibrium differential equation of theplate bending synthesized the force's equilibrium condition, geometric equation and physical equation, and that the boundary conditions of the force and displacement are equivalent. As far as the fixed edge, the simply supported edge and the free edge, there are four parameters on each boundary. Because the unknown quantity number is equaled to the known condition number, the calculating condition is complete. Though the displacement of the column support is known, the reverse force can only be determined by way of the force's equilibrium condition, having nothing to do with the geometric and physical equation. Therefore the calculating condition is complete when the reverse force can be determined by way of the force's equilibrium condition, otherwise is incomplete. The former is called the generalized statically determinate bending, and the latter is called indeterminate bending. The largest merit of this kind of classification method is that the point of column support can be at the any place on the plate. Secondly, the accurate solution rigorously abides by the following rules: the trigonometric series waveform that adopted in the homogeneous solution must suit the deformation form that the boundary conditions can arouse out. Thus the plate supported by a fixed edge or by a simple edge and a column was incorporated. In addition, the accurate solution considers that the equilibrium differential equation of the bending plate is expressed by the deflection parameter. Therefore the all known boundary deflections and normal direction loads should have corresponding particular solutions. The accurate solution adopted composite particular solution, which can all together satisfy the governing differential equation, the deflection along the supported edge, the shear force condition along the free edge, the concentrated force condition at the column support, and raised the precision of the solution.The rectangular plate bending with a supported edge combined the cantilever rectangular plate with the rectangular plate supported by a simple edge and a column, and cancelled the limit that the column must be on the comer point. The solution of this kind of plate made a great difference with the unified solution because of the variety of boundary condition. The very method has the advantage of high precision and has been proved by inverse analysis examples.